{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume6237","volumeNumber":34,"name":"31st International Symposium on Computational Geometry (SoCG 2015)","dateCreated":"2015-06-12","datePublished":"2015-06-12","editor":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6237"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article7844","name":"LIPIcs, Volume 34, SoCG'15, Complete Volume","abstract":"LIPIcs, Volume 34, SoCG'15, Complete Volume","keywords":"Analysis of Algorithms and Problem Complexity, Nonnumerical Algorithms and Problems \u2013 Geometrical problems and computations, Discrete Mathematics","author":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"position":-1,"pageStart":0,"pageEnd":0,"dateCreated":"2015-06-26","datePublished":"2015-06-26","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7845","name":"Front Matter, Table of Contents, Preface, Conference Organization","abstract":"Front Matter, Table of Contents, Preface, Conference Organization","keywords":["Front Matter","Table of Contents","Preface","Conference Organization"],"author":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"position":0,"pageStart":"i","pageEnd":"xx","dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arge, Lars","givenName":"Lars","familyName":"Arge"},{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.i","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7846","name":"Combinatorial Discrepancy for Boxes via the gamma_2 Norm","abstract":"The gamma_2 norm of a real m by n matrix A is the minimum number t such that the column vectors of A are contained in a 0-centered ellipsoid E that in turn is contained in the hypercube [-t, t]^m. This classical quantity is polynomial-time computable and was proved by the second author and Talwar to approximate the hereditary discrepancy: it bounds the hereditary discrepancy from above and from below, up to logarithmic factors. Here we provided a simplified proof of the upper bound and show that both the upper and the lower bound are asymptotically tight in the worst case.\r\n\r\nWe then demonstrate on several examples the power of the gamma_2 norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of log(n)^(d-1) (up to constant factors) for the d-dimensional Tusnady problem, asking for the combinatorial discrepancy of an n-point set in d-dimensional space with respect to axis-parallel boxes. For d>2, this improves the previous best lower bound, which was of order approximately log(n)^((d-1)\/2), and it comes close to the best known upper bound of O(log(n)^(d+1\/2)), for which we also obtain a new, very simple proof. Applications to lower bounds for dynamic range searching and lower bounds in differential privacy are given.","keywords":["discrepancy theory","range counting","factorization norms"],"author":[{"@type":"Person","name":"Matou\u0161ek, Jir\u00ed","givenName":"Jir\u00ed","familyName":"Matou\u0161ek"},{"@type":"Person","name":"Nikolov, Aleksandar","givenName":"Aleksandar","familyName":"Nikolov"}],"position":1,"pageStart":1,"pageEnd":15,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Matou\u0161ek, Jir\u00ed","givenName":"Jir\u00ed","familyName":"Matou\u0161ek"},{"@type":"Person","name":"Nikolov, Aleksandar","givenName":"Aleksandar","familyName":"Nikolov"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/arxiv.org\/abs\/1002.2259","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7847","name":"Tilt: The Video - Designing Worlds to Control Robot Swarms with Only Global Signals","abstract":"We present fundamental progress on the computational universality of swarms of micro- or nano-scale robots in complex environments, controlled not by individual navigation, but by a uniform global, external force. More specifically, we consider a 2D grid world, in which all obstacles and robots are unit squares, and for each actuation, robots move maximally until they collide with an obstacle or another robot. The objective is to control robot motion within obstacles, design obstacles in order to achieve desired permutation of robots, and establish controlled interaction that is complex enough to allow arbitrary computations. In this video, we illustrate progress on all these challenges: we demonstrate NP-hardness of parallel navigation, we describe how to construct obstacles that allow arbitrary permutations, and we establish the necessary logic gates for performing arbitrary in-system computations.","keywords":["Particle swarms","global control","complexity","geometric computation"],"author":[{"@type":"Person","name":"Becker, Aaron T.","givenName":"Aaron T.","familyName":"Becker"},{"@type":"Person","name":"Demaine, Erik D.","givenName":"Erik D.","familyName":"Demaine"},{"@type":"Person","name":"Fekete, S\u00e1ndor P.","givenName":"S\u00e1ndor P.","familyName":"Fekete"},{"@type":"Person","name":"Shad, Hamed Mohtasham","givenName":"Hamed Mohtasham","familyName":"Shad"},{"@type":"Person","name":"Morris-Wright, Rose","givenName":"Rose","familyName":"Morris-Wright"}],"position":2,"pageStart":16,"pageEnd":18,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Becker, Aaron T.","givenName":"Aaron T.","familyName":"Becker"},{"@type":"Person","name":"Demaine, Erik D.","givenName":"Erik D.","familyName":"Demaine"},{"@type":"Person","name":"Fekete, S\u00e1ndor P.","givenName":"S\u00e1ndor P.","familyName":"Fekete"},{"@type":"Person","name":"Shad, Hamed Mohtasham","givenName":"Hamed Mohtasham","familyName":"Shad"},{"@type":"Person","name":"Morris-Wright, Rose","givenName":"Rose","familyName":"Morris-Wright"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/www.thinkfun.com\/tilt","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7848","name":"Automatic Proofs for Formulae Enumerating Proper Polycubes","abstract":"This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions.","keywords":["Polycubes","inclusion-exclusion"],"author":[{"@type":"Person","name":"Barequet, Gill","givenName":"Gill","familyName":"Barequet"},{"@type":"Person","name":"Shalah, Mira","givenName":"Mira","familyName":"Shalah"}],"position":3,"pageStart":19,"pageEnd":22,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Barequet, Gill","givenName":"Gill","familyName":"Barequet"},{"@type":"Person","name":"Shalah, Mira","givenName":"Mira","familyName":"Shalah"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.19","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7849","name":"Visualizing Sparse Filtrations","abstract":"Over the last few years, there have been several approaches to building sparser complexes that still give good approximations to the persistent homology. In this video, we have illustrated a geometric perspective on sparse filtrations that leads to simpler proofs, more general theorems, and a more visual explanation. We hope that as these techniques become easier to understand, they will also become easier to use.","keywords":["Topological Data Analysis","Simplicial Complexes","Persistent Homology"],"author":[{"@type":"Person","name":"Cavanna, Nicholas J.","givenName":"Nicholas J.","familyName":"Cavanna"},{"@type":"Person","name":"Jahanseir, Mahmoodreza","givenName":"Mahmoodreza","familyName":"Jahanseir"},{"@type":"Person","name":"Sheehy, Donald R.","givenName":"Donald R.","familyName":"Sheehy"}],"position":4,"pageStart":23,"pageEnd":25,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cavanna, Nicholas J.","givenName":"Nicholas J.","familyName":"Cavanna"},{"@type":"Person","name":"Jahanseir, Mahmoodreza","givenName":"Mahmoodreza","familyName":"Jahanseir"},{"@type":"Person","name":"Sheehy, Donald R.","givenName":"Donald R.","familyName":"Sheehy"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.23","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7850","name":"Visualizing Quickest Visibility Maps","abstract":"Consider the following modification to the shortest path query problem in polygonal domains: instead of finding shortest path to a query point q, we find the shortest path to any point that sees q. We present an interactive visualization applet visualizing these quickest visibility paths. The applet also visualizes quickest visibility maps, that is the subdivision of the domain into cells by the quickest visibility path structure.","keywords":["path planning","visibility"],"author":{"@type":"Person","name":"Talvitie, Topi","givenName":"Topi","familyName":"Talvitie"},"position":5,"pageStart":26,"pageEnd":28,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Talvitie, Topi","givenName":"Topi","familyName":"Talvitie"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.26","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7851","name":"Sylvester-Gallai for Arrangements of Subspaces","abstract":"In this work we study arrangements of k-dimensional subspaces V_1,...,V_n over the complex numbers. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that the subspaces are pairwise non-intersecting (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. One of the main ingredients in the proof is a strengthening of a theorem of Barthe (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).","keywords":["Sylvester-Gallai","Locally Correctable Codes"],"author":[{"@type":"Person","name":"Dvir, Zeev","givenName":"Zeev","familyName":"Dvir"},{"@type":"Person","name":"Hu, Guangda","givenName":"Guangda","familyName":"Hu"}],"position":6,"pageStart":29,"pageEnd":43,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dvir, Zeev","givenName":"Zeev","familyName":"Dvir"},{"@type":"Person","name":"Hu, Guangda","givenName":"Guangda","familyName":"Hu"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.29","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7852","name":"Computational Aspects of the Colorful Carath\u00e9odory Theorem","abstract":"Let P_1,...,P_{d+1} be d-dimensional point sets such that the convex hull of each P_i contains the origin. We call the sets P_i color classes, and we think of the points in P_i as having color i. A colorful choice is a set with at most one point of each color. The colorful Caratheodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown.\r\n\r\nWe approach this problem from two directions. First, we consider approximation algorithms: an m-colorful choice is a set that contains at most m points from each color class. We show that for any fixed epsilon > 0, an (epsilon d)-colorful choice containing the origin in its convex hull can be found in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Caratheodory theorem in the literature. In the second part, we present a natural generalization of the colorful Caratheodory problem: in the Nearest Colorful Polytope problem (NCP), we are given d-dimensional point sets P_1,...,P_n that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima for the NCP problem is PLS-complete, while computing a global optimum is NP-hard.","keywords":["colorful Carath\u00e9odory theorem","high-dimensional approximation","PLS"],"author":[{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"},{"@type":"Person","name":"Stein, Yannik","givenName":"Yannik","familyName":"Stein"}],"position":7,"pageStart":44,"pageEnd":58,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"},{"@type":"Person","name":"Stein, Yannik","givenName":"Yannik","familyName":"Stein"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.44","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7853","name":"Semi-algebraic Ramsey Numbers","abstract":"Given a finite set P of points from R^d, a k-ary semi-algebraic relation E on P is the set of k-tuples of points in P, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. The Ramsey number R^{d,t}_k(s,n) is the minimum N such that any N-element point set P in R^d equipped with a k-ary semi-algebraic relation E, such that E has complexity at most t, contains s members such that every k-tuple induced by them is in E, or n members such that every k-tuple induced by them is not in E.\r\n\r\nWe give a new upper bound for R^{d,t}_k(s,n) for k=3 and s fixed. In particular, we show that for fixed integers d,t,s, R^{d,t}_3(s,n)=2^{n^{o(1)}}, establishing a subexponential upper bound on R^{d,t}_3(s,n). This improves the previous bound of 2^{n^C} due to Conlon, Fox, Pach, Sudakov, and Suk, where C is a very large constant depending on d,t, and s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in R^d. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.","keywords":["Ramsey theory","semi-algebraic relation","one-sided hyperplanes","Schur numbers"],"author":{"@type":"Person","name":"Suk, Andrew","givenName":"Andrew","familyName":"Suk"},"position":8,"pageStart":59,"pageEnd":73,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Suk, Andrew","givenName":"Andrew","familyName":"Suk"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.59","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7854","name":"A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set","abstract":"In this note it is established that, for any finite set A of real numbers, there exist two elements a, b from A such that |(a + A)(b + A)| > c|A|^2 \/ log |A|, where c is some positive constant. In particular, it follows that |(A + A)(A + A)| > c|A|^2 \/ log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev, which built upon the recent developments of Guth and Katz in their work on the Erd\u00f6s distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemer\u00e9di-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from the paper of the author and Rudnev, since the set (a + A)(b + A) is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product A x A. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when A is a set of complex numbers.","keywords":["Szemer\u00e9di-Trotter Theorem","pinned distances","sum-product estimates"],"author":{"@type":"Person","name":"Roche-Newton, Oliver","givenName":"Oliver","familyName":"Roche-Newton"},"position":9,"pageStart":74,"pageEnd":80,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Roche-Newton, Oliver","givenName":"Oliver","familyName":"Roche-Newton"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.74","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7855","name":"A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes","abstract":"We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.","keywords":["Convex polytopes","Minkowski sum","upper bound"],"author":[{"@type":"Person","name":"Karavelas, Menelaos I.","givenName":"Menelaos I.","familyName":"Karavelas"},{"@type":"Person","name":"Tzanaki, Eleni","givenName":"Eleni","familyName":"Tzanaki"}],"position":10,"pageStart":81,"pageEnd":95,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Karavelas, Menelaos I.","givenName":"Menelaos I.","familyName":"Karavelas"},{"@type":"Person","name":"Tzanaki, Eleni","givenName":"Eleni","familyName":"Tzanaki"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.81","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1405.7368v3","http:\/\/arxiv.org\/abs\/1502.02265v2"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7856","name":"Two Proofs for Shallow Packings","abstract":"We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n \/ delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X' of X of size m <= n and for any parameter 1 <= k <= m, the number of vectors of length at most k in the restriction of V to X' is only O(m^{d_1} k^{d-d_1}), for a fixed integer d > 0 and a real parameter 1 <= d_1 <= d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d). In this case when V is \"k-shallow\" (all vector lengths are at most k), we show that its delta-packing number is O(n^{d_1} k^{d-d_1} \/ delta^d), matching Haussler's bound for the special cases where d_1=d or k=n. We present two proofs, the first is an extension of Haussler's approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler's proof.","keywords":["Set systems of bounded primal shatter dimension","delta-packing & Haussler\u2019s approach","relative approximations","Clarkson-Shor random sampling approach"],"author":[{"@type":"Person","name":"Dutta, Kunal","givenName":"Kunal","familyName":"Dutta"},{"@type":"Person","name":"Ezra, Esther","givenName":"Esther","familyName":"Ezra"},{"@type":"Person","name":"Ghosh, Arijit","givenName":"Arijit","familyName":"Ghosh"}],"position":11,"pageStart":96,"pageEnd":110,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dutta, Kunal","givenName":"Kunal","familyName":"Dutta"},{"@type":"Person","name":"Ezra, Esther","givenName":"Esther","familyName":"Ezra"},{"@type":"Person","name":"Ghosh, Arijit","givenName":"Arijit","familyName":"Ghosh"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.96","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7857","name":"Shortest Path in a Polygon using Sublinear Space","abstract":"We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon P with n vertices in a read only memory, and additional working memory of size m, the new algorithm computes the shortest path (in P) in O(n^2 \/ m) expected time, assuming m = O(n \/ log^2 n). This requires several new tools, which we believe to be of independent interest.\r\n\r\nSpecifically, we show that violator space problems, an abstraction of low dimensional linear-programming (and LP-type problems), can be solved using constant space and expected linear time, by modifying Seidel's linear programming algorithm and using pseudo-random sequences.","keywords":["Shortest path","violator spaces","limited space"],"author":{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},"position":12,"pageStart":111,"pageEnd":125,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.111","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/kam.mff.cuni.cz\/~xofon\/thesis\/diplomka.pdf","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7858","name":"Optimal Morphs of Convex Drawings","abstract":"We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.","keywords":["Convex Drawings","Planar Graphs","Morphing","Geometric Representations"],"author":[{"@type":"Person","name":"Angelini, Patrizio","givenName":"Patrizio","familyName":"Angelini"},{"@type":"Person","name":"Da Lozzo, Giordano","givenName":"Giordano","familyName":"Da Lozzo"},{"@type":"Person","name":"Frati, Fabrizio","givenName":"Fabrizio","familyName":"Frati"},{"@type":"Person","name":"Lubiw, Anna","givenName":"Anna","familyName":"Lubiw"},{"@type":"Person","name":"Patrignani, Maurizio","givenName":"Maurizio","familyName":"Patrignani"},{"@type":"Person","name":"Roselli, Vincenzo","givenName":"Vincenzo","familyName":"Roselli"}],"position":13,"pageStart":126,"pageEnd":140,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Angelini, Patrizio","givenName":"Patrizio","familyName":"Angelini"},{"@type":"Person","name":"Da Lozzo, Giordano","givenName":"Giordano","familyName":"Da Lozzo"},{"@type":"Person","name":"Frati, Fabrizio","givenName":"Fabrizio","familyName":"Frati"},{"@type":"Person","name":"Lubiw, Anna","givenName":"Anna","familyName":"Lubiw"},{"@type":"Person","name":"Patrignani, Maurizio","givenName":"Maurizio","familyName":"Patrignani"},{"@type":"Person","name":"Roselli, Vincenzo","givenName":"Vincenzo","familyName":"Roselli"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.126","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7859","name":"1-String B_2-VPG Representation of Planar Graphs","abstract":"In this paper, we prove that every planar graph has a 1-string B_2-VPG representation - a string representation using paths in a rectangular grid that contain at most two bends. Furthermore, two paths representing vertices u, v intersect precisely once whenever there is an edge between u and v.","keywords":["Graph drawing","string graphs","VPG graphs","planar graphs"],"author":[{"@type":"Person","name":"Biedl, Therese","givenName":"Therese","familyName":"Biedl"},{"@type":"Person","name":"Derka, Martin","givenName":"Martin","familyName":"Derka"}],"position":14,"pageStart":141,"pageEnd":155,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Biedl, Therese","givenName":"Therese","familyName":"Biedl"},{"@type":"Person","name":"Derka, Martin","givenName":"Martin","familyName":"Derka"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.141","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7860","name":"Spanners and Reachability Oracles for Directed Transmission Graphs","abstract":"Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time.\r\n\r\nAfter that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi < sqrt(3), we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Psi >= sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3\/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2\/3)log n) and S(n) = O(n^(5\/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3\/2)) and O(n^(5\/3) log^2 n) in case (ii) and (iii), respectively.","keywords":["Transmission Graphs","Reachability Oracles","Spanner","Intersection Graph"],"author":[{"@type":"Person","name":"Kaplan, Haim","givenName":"Haim","familyName":"Kaplan"},{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"},{"@type":"Person","name":"Roditty, Liam","givenName":"Liam","familyName":"Roditty"},{"@type":"Person","name":"Seiferth, Paul","givenName":"Paul","familyName":"Seiferth"}],"position":15,"pageStart":156,"pageEnd":170,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kaplan, Haim","givenName":"Haim","familyName":"Kaplan"},{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"},{"@type":"Person","name":"Roditty, Liam","givenName":"Liam","familyName":"Roditty"},{"@type":"Person","name":"Seiferth, Paul","givenName":"Paul","familyName":"Seiferth"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.156","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7861","name":"Recognition and Complexity of Point Visibility Graphs","abstract":"A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set.\r\n\r\nWe study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs.\r\n\r\nFurthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.","keywords":["point visibility graphs","recognition","existential theory of the reals"],"author":[{"@type":"Person","name":"Cardinal, Jean","givenName":"Jean","familyName":"Cardinal"},{"@type":"Person","name":"Hoffmann, Udo","givenName":"Udo","familyName":"Hoffmann"}],"position":16,"pageStart":171,"pageEnd":185,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cardinal, Jean","givenName":"Jean","familyName":"Cardinal"},{"@type":"Person","name":"Hoffmann, Udo","givenName":"Udo","familyName":"Hoffmann"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.171","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7862","name":"Geometric Spanners for Points Inside a Polygonal Domain","abstract":"Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function d where for any two points p and q, d(p,q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0), we construct a (sqrt(10)+eps)-spanner that has O(n log^2 n) edges where eps is the a given positive real number. For a case where there are h holes, our construction gives a (5+eps)-spanner with the size of O(sqrt(h) n log^2 n).\r\n \r\nMoreover, we study t-spanners for the visibility graph of P (VG(P), for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+eps)-spanner of size almost O(n^{4\/3}). In addition, we show that there is a set P of n points such that any (3-eps)-spanner of VG(P) must contain almost n^2 edges.","keywords":["Geometric Spanners","Polygonal Domain","Visibility Graph"],"author":[{"@type":"Person","name":"Abam, Mohammad Ali","givenName":"Mohammad Ali","familyName":"Abam"},{"@type":"Person","name":"Adeli, Marjan","givenName":"Marjan","familyName":"Adeli"},{"@type":"Person","name":"Homapour, Hamid","givenName":"Hamid","familyName":"Homapour"},{"@type":"Person","name":"Asadollahpoor, Pooya Zafar","givenName":"Pooya Zafar","familyName":"Asadollahpoor"}],"position":17,"pageStart":186,"pageEnd":197,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Abam, Mohammad Ali","givenName":"Mohammad Ali","familyName":"Abam"},{"@type":"Person","name":"Adeli, Marjan","givenName":"Marjan","familyName":"Adeli"},{"@type":"Person","name":"Homapour, Hamid","givenName":"Hamid","familyName":"Homapour"},{"@type":"Person","name":"Asadollahpoor, Pooya Zafar","givenName":"Pooya Zafar","familyName":"Asadollahpoor"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.186","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7863","name":"An Optimal Algorithm for the Separating Common Tangents of Two Polygons","abstract":"We describe an algorithm for computing the separating common tangents of two simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies to the same side of the line. A separating common tangent of two polygons is a tangent of both polygons where the polygons are lying on different sides of the tangent. Each polygon is given as a read-only array of its corners. If a separating common tangent does not exist, the algorithm reports that. Otherwise, two corners defining a separating common tangent are returned. The algorithm is simple and implies an optimal algorithm for deciding if the convex hulls of two polygons are disjoint or not. This was not known to be possible in linear time and constant workspace prior to this paper.\r\n\r\nAn outer common tangent is a tangent of both polygons where the polygons are on the same side of the tangent. In the case where the convex hulls of the polygons are disjoint, we give an algorithm for computing the outer common tangents in linear time using constant workspace.","keywords":["planar computational geometry","simple polygon","common tangent","optimal algorithm","constant workspace"],"author":{"@type":"Person","name":"Abrahamsen, Mikkel","givenName":"Mikkel","familyName":"Abrahamsen"},"position":18,"pageStart":198,"pageEnd":208,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Abrahamsen, Mikkel","givenName":"Mikkel","familyName":"Abrahamsen"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.198","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7864","name":"A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon","abstract":"Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.","keywords":["Geodesic distance","facility location","1-center problem","simple polygons"],"author":[{"@type":"Person","name":"Ahn, Hee Kap","givenName":"Hee Kap","familyName":"Ahn"},{"@type":"Person","name":"Barba, Luis","givenName":"Luis","familyName":"Barba"},{"@type":"Person","name":"Bose, Prosenjit","givenName":"Prosenjit","familyName":"Bose"},{"@type":"Person","name":"De Carufel, Jean-Lou","givenName":"Jean-Lou","familyName":"De Carufel"},{"@type":"Person","name":"Korman, Matias","givenName":"Matias","familyName":"Korman"},{"@type":"Person","name":"Oh, Eunjin","givenName":"Eunjin","familyName":"Oh"}],"position":19,"pageStart":209,"pageEnd":223,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ahn, Hee Kap","givenName":"Hee Kap","familyName":"Ahn"},{"@type":"Person","name":"Barba, Luis","givenName":"Luis","familyName":"Barba"},{"@type":"Person","name":"Bose, Prosenjit","givenName":"Prosenjit","familyName":"Bose"},{"@type":"Person","name":"De Carufel, Jean-Lou","givenName":"Jean-Lou","familyName":"De Carufel"},{"@type":"Person","name":"Korman, Matias","givenName":"Matias","familyName":"Korman"},{"@type":"Person","name":"Oh, Eunjin","givenName":"Eunjin","familyName":"Oh"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.209","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7865","name":"On the Smoothed Complexity of Convex Hulls","abstract":"We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4\/(d+1)} (1+1\/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2\/3} (1+r^{-2\/3})).\r\n\r\nWe also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise.","keywords":["Probabilistic analysis","Worst-case analysis","Gaussian noise"],"author":[{"@type":"Person","name":"Devillers, Olivier","givenName":"Olivier","familyName":"Devillers"},{"@type":"Person","name":"Glisse, Marc","givenName":"Marc","familyName":"Glisse"},{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Thomasse, R\u00e9my","givenName":"R\u00e9my","familyName":"Thomasse"}],"position":20,"pageStart":224,"pageEnd":239,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Devillers, Olivier","givenName":"Olivier","familyName":"Devillers"},{"@type":"Person","name":"Glisse, Marc","givenName":"Marc","familyName":"Glisse"},{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Thomasse, R\u00e9my","givenName":"R\u00e9my","familyName":"Thomasse"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.224","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7866","name":"Finding All Maximal Subsequences with Hereditary Properties","abstract":"Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results:\r\n\r\n- In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1. \r\n\r\n- In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1.\r\n\r\n- In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction.\r\n\r\nThe same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.","keywords":["convex hull","diameter","monotone path","sequence of points","trajectory"],"author":[{"@type":"Person","name":"Bokal, Drago","givenName":"Drago","familyName":"Bokal"},{"@type":"Person","name":"Cabello, Sergio","givenName":"Sergio","familyName":"Cabello"},{"@type":"Person","name":"Eppstein, David","givenName":"David","familyName":"Eppstein"}],"position":21,"pageStart":240,"pageEnd":254,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bokal, Drago","givenName":"Drago","familyName":"Bokal"},{"@type":"Person","name":"Cabello, Sergio","givenName":"Sergio","familyName":"Cabello"},{"@type":"Person","name":"Eppstein, David","givenName":"David","familyName":"Eppstein"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.240","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7867","name":"Riemannian Simplices and Triangulations","abstract":"We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of the standard simplex under this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the \"thickness\" or \"fatness\" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.","keywords":["Karcher means","barycentric coordinates","triangulation","Riemannian manifold","Riemannian simplices"],"author":[{"@type":"Person","name":"Dyer, Ramsay","givenName":"Ramsay","familyName":"Dyer"},{"@type":"Person","name":"Vegter, Gert","givenName":"Gert","familyName":"Vegter"},{"@type":"Person","name":"Wintraecken, Mathijs","givenName":"Mathijs","familyName":"Wintraecken"}],"position":22,"pageStart":255,"pageEnd":269,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dyer, Ramsay","givenName":"Ramsay","familyName":"Dyer"},{"@type":"Person","name":"Vegter, Gert","givenName":"Gert","familyName":"Vegter"},{"@type":"Person","name":"Wintraecken, Mathijs","givenName":"Mathijs","familyName":"Wintraecken"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.255","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7868","name":"An Edge-Based Framework for Enumerating 3-Manifold Triangulations","abstract":"A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Although censuses are useful resources for mathematicians, constructing them is difficult: the best algorithms to date have not gone beyond n=12. The underlying algorithms essentially (i) enumerate all relevant 4-regular multigraphs on n nodes, and then (ii) for each multigraph G they enumerate possible 3-manifold triangulations with G as their dual 1-skeleton, of which there could be exponentially many. In practice, a small number of multigraphs often dominate the running times of census algorithms: for example, in a typical census on 10 tetrahedra, almost half of the running time is spent on just 0.3% of the graphs.\r\n\r\nHere we present a new algorithm for stage (ii), which is the computational bottleneck in this process. The key idea is to build triangulations by recursively constructing neighbourhoods of edges, in contrast to traditional algorithms which recursively glue together pairs of tetrahedron faces. We implement this algorithm, and find experimentally that whilst the overall performance is mixed, the new algorithm runs significantly faster on those \"pathological\" multigraphs for which existing methods are extremely slow. In this way the old and new algorithms complement one another, and together can yield significant performance improvements over either method alone.","keywords":["triangulations","enumeration","graph theory"],"author":[{"@type":"Person","name":"Burton, Benjamin A.","givenName":"Benjamin A.","familyName":"Burton"},{"@type":"Person","name":"Pettersson, William","givenName":"William","familyName":"Pettersson"}],"position":23,"pageStart":270,"pageEnd":284,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Burton, Benjamin A.","givenName":"Benjamin A.","familyName":"Burton"},{"@type":"Person","name":"Pettersson, William","givenName":"William","familyName":"Pettersson"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.270","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/www.matlas.math.csu.ru\/?page=recognizer","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7869","name":"Order on Order Types","abstract":"Given P and P', equally sized planar point sets in general position, we call a bijection from P to P' crossing-preserving if crossings of connecting segments in P are preserved in P' (extra crossings may occur in P'). If such a mapping exists, we say that P' crossing-dominates P, and if such a mapping exists in both directions, P and P' are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.)\r\n\r\nWe argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.","keywords":["point set","order type","planar graph","crossing-free geometric graph"],"author":[{"@type":"Person","name":"Pilz, Alexander","givenName":"Alexander","familyName":"Pilz"},{"@type":"Person","name":"Welzl, Emo","givenName":"Emo","familyName":"Welzl"}],"position":24,"pageStart":285,"pageEnd":299,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Pilz, Alexander","givenName":"Alexander","familyName":"Pilz"},{"@type":"Person","name":"Welzl, Emo","givenName":"Emo","familyName":"Welzl"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.285","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7870","name":"Limits of Order Types","abstract":"The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs.\r\n\r\nWe first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly.\r\n\r\nWe next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester's problem on the probability that k random points are in convex position.","keywords":["order types","Limits of discrete structures","Flag algebras","Erdos-Szekeres","Sylvester\u2019s problem"],"author":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Hubard, Alfredo","givenName":"Alfredo","familyName":"Hubard"},{"@type":"Person","name":"de Joannis de Verclos, R\u00e9mi","givenName":"R\u00e9mi","familyName":"de Joannis de Verclos"},{"@type":"Person","name":"Sereni, Jean-S\u00e9bastien","givenName":"Jean-S\u00e9bastien","familyName":"Sereni"},{"@type":"Person","name":"Volec, Jan","givenName":"Jan","familyName":"Volec"}],"position":25,"pageStart":300,"pageEnd":314,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Hubard, Alfredo","givenName":"Alfredo","familyName":"Hubard"},{"@type":"Person","name":"de Joannis de Verclos, R\u00e9mi","givenName":"R\u00e9mi","familyName":"de Joannis de Verclos"},{"@type":"Person","name":"Sereni, Jean-S\u00e9bastien","givenName":"Jean-S\u00e9bastien","familyName":"Sereni"},{"@type":"Person","name":"Volec, Jan","givenName":"Jan","familyName":"Volec"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.300","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1002\/rsa.20536","http:\/\/www.sagemath.org"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7871","name":"Combinatorial Redundancy Detection","abstract":"The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs) in dictionary form, given by n equality constraints in n+d variables, where the variables are constrained to be nonnegative. A variable x_r is called redundant, if after removing its nonnegativity constraint the LP still has the same feasible region. The time needed to solve such an LP is denoted by LP(n,d).\r\n\r\nIt is easy to see that solving n+d LPs of the above size is sufficient to detect all redundancies. The currently fastest practical method is the one by Clarkson: it solves n+d linear programs, but each of them has at most s variables, where s is the number of nonredundant constraints.\r\n\r\nIn the first part we show that knowing all of the finitely many dictionaries of the LP is sufficient for the purpose of redundancy detection. A dictionary is a matrix that can be thought of as an enriched encoding of a vertex in the LP. Moreover - and this is the combinatorial aspect - it is enough to know only the signs of the entries, the actual values do not matter. Concretely we show that for any variable x_r one can find a dictionary, such that its sign pattern is either a redundancy or nonredundancy certificate for x_r. \r\n\r\nIn the second part we show that considering only the sign patterns of the dictionary, there is an output sensitive algorithm of running time of order d (n+d) s^{d-1} LP(s,d) + d s^{d} LP(n,d) to detect all redundancies. In the case where all constraints are in general position, the running time is of order s LP(n,d) + (n+d) LP(s,d), which is essentially the running time of the Clarkson method. Our algorithm extends naturally to a more general setting of arrangements of oriented topological hyperplane arrangements.","keywords":["system of linear inequalities","redundancy removal","linear programming","output sensitive algorithm","Clarkson\u2019s method"],"author":[{"@type":"Person","name":"Fukuda, Komei","givenName":"Komei","familyName":"Fukuda"},{"@type":"Person","name":"G\u00e4rtner, Bernd","givenName":"Bernd","familyName":"G\u00e4rtner"},{"@type":"Person","name":"Szedl\u00e1k, May","givenName":"May","familyName":"Szedl\u00e1k"}],"position":26,"pageStart":315,"pageEnd":328,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Fukuda, Komei","givenName":"Komei","familyName":"Fukuda"},{"@type":"Person","name":"G\u00e4rtner, Bernd","givenName":"Bernd","familyName":"G\u00e4rtner"},{"@type":"Person","name":"Szedl\u00e1k, May","givenName":"May","familyName":"Szedl\u00e1k"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.315","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/www.ifor.math.ethz.ch\/teaching\/Courses\/Fall_2011\/intro_fall_11","http:\/\/helper.ipam.ucla.edu\/publications\/sm2011\/sm2011_9630.pdf","http:\/\/www-oldurls.inf.ethz.ch\/personal\/fukudak\/lect\/pclect\/notes2015\/"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7872","name":"Effectiveness of Local Search for Geometric Optimization","abstract":"What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude 1\/epsilon^c is an approximation scheme for the following problems in the Euclidean plane: TSP with random inputs, Steiner tree with random inputs, uniform facility location (with worst case inputs), and bicriteria k-median (also with worst case inputs). The randomness assumption is necessary for TSP.","keywords":["Local Search","PTAS","Facility Location","k-Median","TSP","Steiner Tree"],"author":[{"@type":"Person","name":"Cohen-Addad, Vincent","givenName":"Vincent","familyName":"Cohen-Addad"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"}],"position":27,"pageStart":329,"pageEnd":344,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cohen-Addad, Vincent","givenName":"Vincent","familyName":"Cohen-Addad"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.329","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7873","name":"On the Shadow Simplex Method for Curved Polyhedra","abstract":"We study the simplex method over polyhedra satisfying certain \"discrete curvature\" lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (SOCG 2012), Brunsch and R\u00f6glin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved our understanding of such polyhedra.\r\n\r\nWe develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author, who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing.\r\n\r\nFor our first result, we obtain a constructive diameter bound of O((n^2 \/ delta) ln (n \/ delta)) for n-dimensional polyhedra with curvature parameter delta in (0, 1]. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of O(n^3 ln n). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected O((n^3 \/ delta) ln (n \/ delta)) simplex pivots, each requiring O(mn) time to compute. An initial feasible solution can be found using O((mn^3 \/ delta) ln (n \/ delta)) pivot steps.","keywords":["Optimization","Linear Programming","Simplex Method","Diameter of Polyhedra"],"author":[{"@type":"Person","name":"Dadush, Daniel","givenName":"Daniel","familyName":"Dadush"},{"@type":"Person","name":"H\u00e4hnle, Nicolai","givenName":"Nicolai","familyName":"H\u00e4hnle"}],"position":28,"pageStart":345,"pageEnd":359,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dadush, Daniel","givenName":"Daniel","familyName":"Dadush"},{"@type":"Person","name":"H\u00e4hnle, Nicolai","givenName":"Nicolai","familyName":"H\u00e4hnle"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.345","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7874","name":"Pattern Overlap Implies Runaway Growth in Hierarchical Tile Systems","abstract":"We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap.\r\n\r\nThis answers an open question of Chen and Doty (SODA 2012), who showed that so-called \"partial-order\" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.","keywords":["self-assembly","hierarchical","pumping"],"author":[{"@type":"Person","name":"Chen, Ho-Lin","givenName":"Ho-Lin","familyName":"Chen"},{"@type":"Person","name":"Doty, David","givenName":"David","familyName":"Doty"},{"@type":"Person","name":"Manuch, J\u00e1n","givenName":"J\u00e1n","familyName":"Manuch"},{"@type":"Person","name":"Rafiey, Arash","givenName":"Arash","familyName":"Rafiey"},{"@type":"Person","name":"Stacho, Ladislav","givenName":"Ladislav","familyName":"Stacho"}],"position":29,"pageStart":360,"pageEnd":373,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chen, Ho-Lin","givenName":"Ho-Lin","familyName":"Chen"},{"@type":"Person","name":"Doty, David","givenName":"David","familyName":"Doty"},{"@type":"Person","name":"Manuch, J\u00e1n","givenName":"J\u00e1n","familyName":"Manuch"},{"@type":"Person","name":"Rafiey, Arash","givenName":"Arash","familyName":"Rafiey"},{"@type":"Person","name":"Stacho, Ladislav","givenName":"Ladislav","familyName":"Stacho"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.360","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7875","name":"Space Exploration via Proximity Search","abstract":"We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following:\r\n\r\n(A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set.\r\n\r\n(B) One can decide if a query point is (approximately) inside the convex-hull of the point set.\r\n\r\nWe also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.","keywords":["Proximity search","implicit point set","probing"],"author":[{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},{"@type":"Person","name":"Kumar, Nirman","givenName":"Nirman","familyName":"Kumar"},{"@type":"Person","name":"Mount, David M.","givenName":"David M.","familyName":"Mount"},{"@type":"Person","name":"Raichel, Benjamin","givenName":"Benjamin","familyName":"Raichel"}],"position":30,"pageStart":374,"pageEnd":389,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},{"@type":"Person","name":"Kumar, Nirman","givenName":"Nirman","familyName":"Kumar"},{"@type":"Person","name":"Mount, David M.","givenName":"David M.","familyName":"Mount"},{"@type":"Person","name":"Raichel, Benjamin","givenName":"Benjamin","familyName":"Raichel"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.374","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7876","name":"Star Unfolding from a Geodesic Curve","abstract":"There are two known ways to unfold a convex polyhedron without overlap: the star unfolding and the source unfolding, both of which use shortest paths from vertices to a source point on the surface of the polyhedron. Non-overlap of the source unfolding is straightforward; non-overlap of the star unfolding was proved by Aronov and O'Rourke in 1992. Our first contribution is a much simpler proof of non-overlap of the star unfolding.\r\n\r\nBoth the source and star unfolding can be generalized to use a simple geodesic curve instead of a source point. The star unfolding from a geodesic curve cuts the geodesic curve and a shortest path from each vertex to the geodesic curve. Demaine and Lubiw conjectured that the star unfolding from a geodesic curve does not overlap. We prove a special case of the conjecture. Our special case includes the previously known case of unfolding from a geodesic loop. For the general case we prove that the star unfolding from a geodesic curve can be separated into at most two non-overlapping pieces.","keywords":["unfolding","convex polyhedra","geodesic curve"],"author":[{"@type":"Person","name":"Kiazyk, Stephen","givenName":"Stephen","familyName":"Kiazyk"},{"@type":"Person","name":"Lubiw, Anna","givenName":"Anna","familyName":"Lubiw"}],"position":31,"pageStart":390,"pageEnd":404,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kiazyk, Stephen","givenName":"Stephen","familyName":"Kiazyk"},{"@type":"Person","name":"Lubiw, Anna","givenName":"Anna","familyName":"Lubiw"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.390","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7877","name":"The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)","abstract":"Suppose you have n points in the plane, not all on a line. A famous theorem of Sylvester-Gallai asserts that there is at least one ordinary line, that is to say a line passing through precisely two of the n points. But how many ordinary lines must there be? It turns out that the answer is at least n\/2 (if n is even) and roughly 3n\/4 (if n is odd), provided that n is sufficiently large. This resolves a conjecture of Dirac and Motzkin from the 1950s. We will also discuss the classical orchard problem, which asks how to arrange n trees so that there are as many triples of colinear trees as possible, but no four in a line. This is joint work with Terence Tao and reports on the results of [Green and Tao, 2013].","keywords":["combinatorial geometry","incidences"],"author":{"@type":"Person","name":"Green, Ben J.","givenName":"Ben J.","familyName":"Green"},"position":32,"pageStart":405,"pageEnd":405,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Green, Ben J.","givenName":"Ben J.","familyName":"Green"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.405","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7878","name":"On the Beer Index of Convexity and Its Variants","abstract":"Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S.\r\n\r\nWe show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons.\r\n\r\nWe also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)\/log_2(1\/epsilon) >= (gamma c(S))\/log_2(1\/c(S)).","keywords":["Beer index of convexity","convexity ratio","convexity measure","visibility"],"author":[{"@type":"Person","name":"Balko, Martin","givenName":"Martin","familyName":"Balko"},{"@type":"Person","name":"Jel\u00ednek, V\u00edt","givenName":"V\u00edt","familyName":"Jel\u00ednek"},{"@type":"Person","name":"Valtr, Pavel","givenName":"Pavel","familyName":"Valtr"},{"@type":"Person","name":"Walczak, Bartosz","givenName":"Bartosz","familyName":"Walczak"}],"position":33,"pageStart":406,"pageEnd":420,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Balko, Martin","givenName":"Martin","familyName":"Balko"},{"@type":"Person","name":"Jel\u00ednek, V\u00edt","givenName":"V\u00edt","familyName":"Jel\u00ednek"},{"@type":"Person","name":"Valtr, Pavel","givenName":"Pavel","familyName":"Valtr"},{"@type":"Person","name":"Walczak, Bartosz","givenName":"Bartosz","familyName":"Walczak"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.406","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/arxiv.org\/abs\/1412.1769","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7879","name":"Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries","abstract":"The chromatic art gallery problem asks for the minimum number of \"colors\" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n \/ logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.","keywords":["Orthogonal polygons","art gallery problem","hypergraph coloring"],"author":[{"@type":"Person","name":"Hoffmann, Frank","givenName":"Frank","familyName":"Hoffmann"},{"@type":"Person","name":"Kriegel, Klaus","givenName":"Klaus","familyName":"Kriegel"},{"@type":"Person","name":"Suri, Subhash","givenName":"Subhash","familyName":"Suri"},{"@type":"Person","name":"Verbeek, Kevin","givenName":"Kevin","familyName":"Verbeek"},{"@type":"Person","name":"Willert, Max","givenName":"Max","familyName":"Willert"}],"position":34,"pageStart":421,"pageEnd":435,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Hoffmann, Frank","givenName":"Frank","familyName":"Hoffmann"},{"@type":"Person","name":"Kriegel, Klaus","givenName":"Klaus","familyName":"Kriegel"},{"@type":"Person","name":"Suri, Subhash","givenName":"Subhash","familyName":"Suri"},{"@type":"Person","name":"Verbeek, Kevin","givenName":"Kevin","familyName":"Verbeek"},{"@type":"Person","name":"Willert, Max","givenName":"Max","familyName":"Willert"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.421","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7880","name":"Low-Quality Dimension Reduction and High-Dimensional Approximate Nearest Neighbor","abstract":"The approximate nearest neighbor problem (epsilon-ANN) in Euclidean settings is a fundamental question, which has been addressed by two main approaches: Data-dependent space partitioning techniques perform well when the dimension is relatively low, but are affected by the curse of dimensionality. On the other hand, locality sensitive hashing has polynomial dependence in the dimension, sublinear query time with an exponent inversely proportional to (1+epsilon)^2, and subquadratic space requirement. \r\n\r\nWe generalize the Johnson-Lindenstrauss Lemma to define \"low-quality\" mappings to a Euclidean space of significantly lower dimension, such that they satisfy a requirement weaker than approximately preserving all distances or even preserving the nearest neighbor. This mapping guarantees, with high probability, that an approximate nearest neighbor lies among the k approximate nearest neighbors in the projected space. These can be efficiently retrieved while using only linear storage by a data structure, such as BBD-trees. Our overall algorithm, given n points in dimension d, achieves space usage in O(dn), preprocessing time in O(dn log n), and query time in O(d n^{rho} log n), where rho is proportional to 1 - 1\/loglog n, for fixed epsilon in (0, 1). The dimension reduction is larger if one assumes that point sets possess some structure, namely bounded expansion rate. We implement our method and present experimental results in up to 500 dimensions and 10^6 points, which show that the practical performance is better than predicted by the theoretical analysis. In addition, we compare our approach with E2LSH.","keywords":["Approximate nearest neighbor","Randomized embeddings","Curse of dimensionality","Johnson-Lindenstrauss Lemma","Bounded expansion rate","Experimental study"],"author":[{"@type":"Person","name":"Anagnostopoulos, Evangelos","givenName":"Evangelos","familyName":"Anagnostopoulos"},{"@type":"Person","name":"Emiris, Ioannis Z.","givenName":"Ioannis Z.","familyName":"Emiris"},{"@type":"Person","name":"Psarros, Ioannis","givenName":"Ioannis","familyName":"Psarros"}],"position":35,"pageStart":436,"pageEnd":450,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Anagnostopoulos, Evangelos","givenName":"Evangelos","familyName":"Anagnostopoulos"},{"@type":"Person","name":"Emiris, Ioannis Z.","givenName":"Ioannis Z.","familyName":"Emiris"},{"@type":"Person","name":"Psarros, Ioannis","givenName":"Ioannis","familyName":"Psarros"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.436","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/www.mit.edu\/~andoni\/LSH","http:\/\/www.cs.umd.edu\/~mount\/ANN\/"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7881","name":"Restricted Isometry Property for General p-Norms","abstract":"The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an m x n matrix satisfies RIP of order k for the L_p norm, if |Ax|_p is approximately |x|_p for every x with at most k non-zero coordinates.\r\n\r\nFor every 1 <= p < infty we obtain almost tight bounds on the minimum number of rows m necessary for the RIP property to hold. Prior to this work, only the cases p = 1, 1 + 1\/log(k), and 2 were studied. Interestingly, our results show that the case p=2 is a \"singularity\" point: the optimal number of rows m is Theta(k^p) for all p in [1, infty)-{2}, as opposed to Theta(k) for k=2.\r\n\r\nWe also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.","keywords":["compressive sensing","dimension reduction","linear algebra","high-dimensional geometry"],"author":[{"@type":"Person","name":"Allen-Zhu, Zeyuan","givenName":"Zeyuan","familyName":"Allen-Zhu"},{"@type":"Person","name":"Gelashvili, Rati","givenName":"Rati","familyName":"Gelashvili"},{"@type":"Person","name":"Razenshteyn, Ilya","givenName":"Ilya","familyName":"Razenshteyn"}],"position":36,"pageStart":451,"pageEnd":460,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Allen-Zhu, Zeyuan","givenName":"Zeyuan","familyName":"Allen-Zhu"},{"@type":"Person","name":"Gelashvili, Rati","givenName":"Rati","familyName":"Gelashvili"},{"@type":"Person","name":"Razenshteyn, Ilya","givenName":"Ilya","familyName":"Razenshteyn"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.451","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7882","name":"Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs","abstract":"The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov-Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.","keywords":["Reeb graph","interleaving distance","functional distortion distance"],"author":[{"@type":"Person","name":"Bauer, Ulrich","givenName":"Ulrich","familyName":"Bauer"},{"@type":"Person","name":"Munch, Elizabeth","givenName":"Elizabeth","familyName":"Munch"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"position":37,"pageStart":461,"pageEnd":475,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bauer, Ulrich","givenName":"Ulrich","familyName":"Bauer"},{"@type":"Person","name":"Munch, Elizabeth","givenName":"Elizabeth","familyName":"Munch"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.461","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/arxiv.org\/abs\/1411.1544","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7883","name":"On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result","abstract":"The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2.\r\n\r\nTwo decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem.\r\n\r\nIn the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.","keywords":["Heawood Inequality","Embeddings","Van Kampen\u2013Flores","Manifolds"],"author":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Mabillard, Isaac","givenName":"Isaac","familyName":"Mabillard"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"position":38,"pageStart":476,"pageEnd":490,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Mabillard, Isaac","givenName":"Isaac","familyName":"Mabillard"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.476","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7884","name":"Comparing Graphs via Persistence Distortion","abstract":"Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the non-linear structure hidden behind the data. In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence-distortion distance has two properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against, for example, different discretizations of the same underlying graph.\r\n\r\nDespite considering the input graphs as continuous spaces, that is, taking all points into account, we show that we can compute the persistence-distortion distance in polynomial time. The time complexity for the discrete case where only graph nodes are considered is much faster.","keywords":["Graph matching","metric graphs","persistence distortion","topological method"],"author":[{"@type":"Person","name":"Dey, Tamal K.","givenName":"Tamal K.","familyName":"Dey"},{"@type":"Person","name":"Shi, Dayu","givenName":"Dayu","familyName":"Shi"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"position":39,"pageStart":491,"pageEnd":506,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dey, Tamal K.","givenName":"Tamal K.","familyName":"Dey"},{"@type":"Person","name":"Shi, Dayu","givenName":"Dayu","familyName":"Shi"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.491","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7885","name":"Bounding Helly Numbers via Betti Numbers","abstract":"We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)\/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.\r\n\r\nOur proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.","keywords":["Helly-type theorem","Ramsey\u2019s theorem","Embedding of simplicial complexes","Homological almost-embedding","Betti numbers"],"author":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"position":40,"pageStart":507,"pageEnd":521,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Goaoc, Xavier","givenName":"Xavier","familyName":"Goaoc"},{"@type":"Person","name":"Pat\u00e1k, Pavel","givenName":"Pavel","familyName":"Pat\u00e1k"},{"@type":"Person","name":"Pat\u00e1kov\u00e1, Zuzana","givenName":"Zuzana","familyName":"Pat\u00e1kov\u00e1"},{"@type":"Person","name":"Tancer, Martin","givenName":"Martin","familyName":"Tancer"},{"@type":"Person","name":"Wagner, Uli","givenName":"Uli","familyName":"Wagner"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.507","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1101.6006","http:\/\/arxiv.org\/abs\/1310.4613"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7886","name":"Polynomials Vanishing on Cartesian Products: The Elekes-Szab\u00f3 Theorem Revisited","abstract":"Let F in Complex[x,y,z] be a constant-degree polynomial, and let A,B,C be sets of complex numbers with |A|=|B|=|C|=n. We show that F vanishes on at most O(n^{11\/6}) points of the Cartesian product A x B x C (where the constant of proportionality depends polynomially on the degree of F), unless F has a special group-related form. This improves a theorem of Elekes and Szabo [ES12], and generalizes a result of Raz, Sharir, and Solymosi [RSS14a]. The same statement holds over R. When A, B, C have different sizes, a similar statement holds, with a more involved bound replacing O(n^{11\/6}).\r\n\r\nThis result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.","keywords":["Combinatorial geometry","incidences","polynomials"],"author":[{"@type":"Person","name":"Raz, Orit E.","givenName":"Orit E.","familyName":"Raz"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"},{"@type":"Person","name":"de Zeeuw, Frank","givenName":"Frank","familyName":"de Zeeuw"}],"position":41,"pageStart":522,"pageEnd":536,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Raz, Orit E.","givenName":"Orit E.","familyName":"Raz"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"},{"@type":"Person","name":"de Zeeuw, Frank","givenName":"Frank","familyName":"de Zeeuw"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.522","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7887","name":"Bisector Energy and Few Distinct Distances","abstract":"We introduce the bisector energy of an n-point set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2\/5}n^{12\/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results:\r\n\r\n(i) If P determines O(n \/ sqrt(log n)) distinct distances, then for any 0 < a < 1\/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8\/5 - 12a\/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of Erd\u00f6s regarding the structure of point sets with few distinct distances.\r\n\r\n(ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{-2\/5}n^{8\/5}, M(n)^{-1}n^2}). This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.","keywords":["Combinatorial geometry","distinct distances","incidence geometry"],"author":[{"@type":"Person","name":"Lund, Ben","givenName":"Ben","familyName":"Lund"},{"@type":"Person","name":"Sheffer, Adam","givenName":"Adam","familyName":"Sheffer"},{"@type":"Person","name":"de Zeeuw, Frank","givenName":"Frank","familyName":"de Zeeuw"}],"position":42,"pageStart":537,"pageEnd":552,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Lund, Ben","givenName":"Ben","familyName":"Lund"},{"@type":"Person","name":"Sheffer, Adam","givenName":"Adam","familyName":"Sheffer"},{"@type":"Person","name":"de Zeeuw, Frank","givenName":"Frank","familyName":"de Zeeuw"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.537","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7888","name":"Incidences between Points and Lines in Three Dimensions","abstract":"We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R^3, so that no plane contains more than s lines, is O(m^{1\/2}n^{3\/4} + m^{2\/3}n^{1\/3}s^{1\/3} + m + n) (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n).\r\n\r\nThis bound, originally obtained by Guth and Katz as a major step in their solution of Erdos's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth. The present paper presents a different and simpler derivation, with better bounds than those in Guth, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.","keywords":["Combinatorial Geometry","Algebraic Geometry","Incidences","The Polynomial Method"],"author":[{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"},{"@type":"Person","name":"Solomon, Noam","givenName":"Noam","familyName":"Solomon"}],"position":43,"pageStart":553,"pageEnd":568,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"},{"@type":"Person","name":"Solomon, Noam","givenName":"Noam","familyName":"Solomon"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.553","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7889","name":"The Number of Unit-Area Triangles in the Plane: Theme and Variations","abstract":"We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n^{20\/9}), improving the earlier bound O(n^{9\/4}) of Apfelbaum and Sharir. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be Omega(n^2), for any triple of lines (it is always O(n^2) in this case). (ii) We show that if S is a convex grid of the form A x B, where A, B are convex sets of n^{1\/2} real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n^{31\/14}) unit-area triangles.","keywords":["Combinatorial geometry","incidences","repeated configurations"],"author":[{"@type":"Person","name":"Raz, Orit E.","givenName":"Orit E.","familyName":"Raz"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"}],"position":44,"pageStart":569,"pageEnd":583,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Raz, Orit E.","givenName":"Orit E.","familyName":"Raz"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.569","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7890","name":"On the Number of Rich Lines in Truly High Dimensional Sets","abstract":"We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a 'truly' d-dimensional configuration of points v_1,...,v_n over the complex numbers. More formally, we show that, if the number of r-rich lines is significantly larger than n^2\/r^d then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^d can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of n^2\/r^3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R^3 in the seminal work of Guth and Katz and was also recently proved over R^4 (under some additional restrictions) by Solomon and Sharir. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions.\r\n\r\nThe main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r-2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of Barak et al. to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.","keywords":["Incidences","Combinatorial Geometry","Designs","Polynomial Method","Additive Combinatorics"],"author":[{"@type":"Person","name":"Dvir, Zeev","givenName":"Zeev","familyName":"Dvir"},{"@type":"Person","name":"Gopi, Sivakanth","givenName":"Sivakanth","familyName":"Gopi"}],"position":45,"pageStart":584,"pageEnd":598,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dvir, Zeev","givenName":"Zeev","familyName":"Dvir"},{"@type":"Person","name":"Gopi, Sivakanth","givenName":"Sivakanth","familyName":"Gopi"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.584","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7891","name":"Realization Spaces of Arrangements of Convex Bodies","abstract":"We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. On one hand, we show that every combinatorial type can be realized by an arrangement of convex bodies and (under mild assumptions) its realization space is contractible. On the other hand, we prove a universality theorem that says that the restriction of the realization space to arrangements of convex polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.","keywords":["Oriented matroids","Convex sets","Realization spaces","Mnev\u2019s universality theorem"],"author":[{"@type":"Person","name":"Dobbins, Michael Gene","givenName":"Michael Gene","familyName":"Dobbins"},{"@type":"Person","name":"Holmsen, Andreas","givenName":"Andreas","familyName":"Holmsen"},{"@type":"Person","name":"Hubard, Alfredo","givenName":"Alfredo","familyName":"Hubard"}],"position":46,"pageStart":599,"pageEnd":614,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dobbins, Michael Gene","givenName":"Michael Gene","familyName":"Dobbins"},{"@type":"Person","name":"Holmsen, Andreas","givenName":"Andreas","familyName":"Holmsen"},{"@type":"Person","name":"Hubard, Alfredo","givenName":"Alfredo","familyName":"Hubard"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.599","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7892","name":"Computing Teichm\u00fcller Maps between Polygons","abstract":"By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle preserving manner). However, when this map is extended to the boundary it need not necessarily map the vertices of P to those of Q. For many applications it is important to find the \"best\" vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichm\u00fcller maps.\r\n\r\nThere are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a (1+epsilon)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichm\u00fcller maps.\r\n\r\nWe present two results in this paper. One studies the problem in the continuous setting and another in the discrete setting.\r\n\r\nIn the continuous setting, we solve the problem of finding a finite time procedure for approximating Teichm\u00fcller maps. Our construction is via an iterative procedure that is proven to converge in O(poly(1\/epsilon)) iterations to a (1+epsilon)-approximation of the Teichmuller map. Our method uses a reduction of the polygon mapping problem to the marked sphere problem, thus solving a more general problem.\r\n\r\nIn the discrete setting, we reduce the problem of finding an approximation algorithm for computing Teichm\u00fcller maps to two basic subroutines, namely, computing discrete 1) compositions and 2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines we provide a (1+epsilon)-approximation algorithm.","keywords":["Teichm\u00fcller maps","Surface registration","Extremal Quasiconformal maps","Computer vision"],"author":[{"@type":"Person","name":"Goswami, Mayank","givenName":"Mayank","familyName":"Goswami"},{"@type":"Person","name":"Gu, Xianfeng","givenName":"Xianfeng","familyName":"Gu"},{"@type":"Person","name":"Pingali, Vamsi P.","givenName":"Vamsi P.","familyName":"Pingali"},{"@type":"Person","name":"Telang, Gaurish","givenName":"Gaurish","familyName":"Telang"}],"position":47,"pageStart":615,"pageEnd":629,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Goswami, Mayank","givenName":"Mayank","familyName":"Goswami"},{"@type":"Person","name":"Gu, Xianfeng","givenName":"Xianfeng","familyName":"Gu"},{"@type":"Person","name":"Pingali, Vamsi P.","givenName":"Vamsi P.","familyName":"Pingali"},{"@type":"Person","name":"Telang, Gaurish","givenName":"Gaurish","familyName":"Telang"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.615","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/www.math.sunysb.edu\/~bishop\/vita\/nsf12.pdf","http:\/\/arxiv.org\/abs\/1401.6395","http:\/\/cis.jhu.edu\/software\/lddmm-volume\/tutorial.php","http:\/\/arxiv.org\/abs\/1307.2679"],"isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7893","name":"On-line Coloring between Two Lines","abstract":"We study on-line colorings of certain graphs given as intersection graphs of objects \"between two lines\", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(w^3) colors on graphs with maximum clique size w.\r\n\r\nIn contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when w=2.\r\n\r\nThe left-of relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class C of posets obtained from convex sets between two lines with some other classes of posets: all 2-dimensional posets and all posets of height 2 are in C but there is a 3-dimensional poset of height 3 that does not belong to C.\r\n\r\nWe also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.","keywords":["intersection graphs","cocomparability graphs","on-line coloring"],"author":[{"@type":"Person","name":"Felsner, Stefan","givenName":"Stefan","familyName":"Felsner"},{"@type":"Person","name":"Micek, Piotr","givenName":"Piotr","familyName":"Micek"},{"@type":"Person","name":"Ueckerdt, Torsten","givenName":"Torsten","familyName":"Ueckerdt"}],"position":48,"pageStart":630,"pageEnd":641,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Felsner, Stefan","givenName":"Stefan","familyName":"Felsner"},{"@type":"Person","name":"Micek, Piotr","givenName":"Piotr","familyName":"Micek"},{"@type":"Person","name":"Ueckerdt, Torsten","givenName":"Torsten","familyName":"Ueckerdt"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.630","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7894","name":"Building Efficient and Compact Data Structures for Simplicial Complexes","abstract":"The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called Maximal Simplex Tree (MxST) and Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.","keywords":["Simplicial complex","Compact data structures","Automaton","NP-hard"],"author":[{"@type":"Person","name":"Boissonnat, Jean-Daniel","givenName":"Jean-Daniel","familyName":"Boissonnat"},{"@type":"Person","name":"S., Karthik C.","givenName":"Karthik C.","familyName":"S."},{"@type":"Person","name":"Tavenas, S\u00e9bastien","givenName":"S\u00e9bastien","familyName":"Tavenas"}],"position":49,"pageStart":642,"pageEnd":657,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Boissonnat, Jean-Daniel","givenName":"Jean-Daniel","familyName":"Boissonnat"},{"@type":"Person","name":"S., Karthik C.","givenName":"Karthik C.","familyName":"S."},{"@type":"Person","name":"Tavenas, S\u00e9bastien","givenName":"S\u00e9bastien","familyName":"Tavenas"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.642","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7895","name":"Shortest Path to a Segment and Quickest Visibility Queries","abstract":"We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.","keywords":["path planning","visibility","query structures and complexity","persistent data structures","continuous Dijkstra"],"author":[{"@type":"Person","name":"Arkin, Esther M.","givenName":"Esther M.","familyName":"Arkin"},{"@type":"Person","name":"Efrat, Alon","givenName":"Alon","familyName":"Efrat"},{"@type":"Person","name":"Knauer, Christian","givenName":"Christian","familyName":"Knauer"},{"@type":"Person","name":"Mitchell, Joseph S. B.","givenName":"Joseph S. B.","familyName":"Mitchell"},{"@type":"Person","name":"Polishchuk, Valentin","givenName":"Valentin","familyName":"Polishchuk"},{"@type":"Person","name":"Rote, G\u00fcnter","givenName":"G\u00fcnter","familyName":"Rote"},{"@type":"Person","name":"Schlipf, Lena","givenName":"Lena","familyName":"Schlipf"},{"@type":"Person","name":"Talvitie, Topi","givenName":"Topi","familyName":"Talvitie"}],"position":50,"pageStart":658,"pageEnd":673,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arkin, Esther M.","givenName":"Esther M.","familyName":"Arkin"},{"@type":"Person","name":"Efrat, Alon","givenName":"Alon","familyName":"Efrat"},{"@type":"Person","name":"Knauer, Christian","givenName":"Christian","familyName":"Knauer"},{"@type":"Person","name":"Mitchell, Joseph S. B.","givenName":"Joseph S. B.","familyName":"Mitchell"},{"@type":"Person","name":"Polishchuk, Valentin","givenName":"Valentin","familyName":"Polishchuk"},{"@type":"Person","name":"Rote, G\u00fcnter","givenName":"G\u00fcnter","familyName":"Rote"},{"@type":"Person","name":"Schlipf, Lena","givenName":"Lena","familyName":"Schlipf"},{"@type":"Person","name":"Talvitie, Topi","givenName":"Topi","familyName":"Talvitie"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.658","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/www.cgal.org","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7896","name":"Trajectory Grouping Structure under Geodesic Distance","abstract":"In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure.\r\n\r\nSignificantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly epsilon, where epsilon is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with tau vertices each we give an O(tau n^2) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(tau(n^2 + m lambda_4(n))) upper bound, where m is the total complexity of the obstacles, and lambda_s(n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(tau min(n^2 + m^3 lambda_4(n), n^2m^2)) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.","keywords":["moving entities","trajectories","grouping","computational geometry"],"author":[{"@type":"Person","name":"Kostitsyna, Irina","givenName":"Irina","familyName":"Kostitsyna"},{"@type":"Person","name":"van Kreveld, Marc","givenName":"Marc","familyName":"van Kreveld"},{"@type":"Person","name":"L\u00f6ffler, Maarten","givenName":"Maarten","familyName":"L\u00f6ffler"},{"@type":"Person","name":"Speckmann, Bettina","givenName":"Bettina","familyName":"Speckmann"},{"@type":"Person","name":"Staals, Frank","givenName":"Frank","familyName":"Staals"}],"position":51,"pageStart":674,"pageEnd":688,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kostitsyna, Irina","givenName":"Irina","familyName":"Kostitsyna"},{"@type":"Person","name":"van Kreveld, Marc","givenName":"Marc","familyName":"van Kreveld"},{"@type":"Person","name":"L\u00f6ffler, Maarten","givenName":"Maarten","familyName":"L\u00f6ffler"},{"@type":"Person","name":"Speckmann, Bettina","givenName":"Bettina","familyName":"Speckmann"},{"@type":"Person","name":"Staals, Frank","givenName":"Frank","familyName":"Staals"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.674","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7897","name":"From Proximity to Utility: A Voronoi Partition of Pareto Optima","abstract":"We present an extension of Voronoi diagrams where not only the distance to the site is taken into account when considering which site the client is going to use, but additional attributes (i.e., prices or weights) are also considered. A cell in this diagram is then the loci of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest.","keywords":["Voronoi diagrams","expected complexity","backward analysis","Pareto optima","candidate diagram","Clarkson-Shor technique"],"author":[{"@type":"Person","name":"Chang, Hsien-Chih","givenName":"Hsien-Chih","familyName":"Chang"},{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},{"@type":"Person","name":"Raichel, Benjamin","givenName":"Benjamin","familyName":"Raichel"}],"position":52,"pageStart":689,"pageEnd":703,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chang, Hsien-Chih","givenName":"Hsien-Chih","familyName":"Chang"},{"@type":"Person","name":"Har-Peled, Sariel","givenName":"Sariel","familyName":"Har-Peled"},{"@type":"Person","name":"Raichel, Benjamin","givenName":"Benjamin","familyName":"Raichel"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.689","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7898","name":"Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings","abstract":"We give a 2^{O(n)}(1+1\/eps)^n time and poly(n)-space deterministic algorithm for computing a (1+eps)^n approximation to the volume of a general convex body K, which comes close to matching the (1+c\/eps)^{n\/2} lower bound for volume estimation in the oracle model by Barany and Furedi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat'l Acad. Sci. 2013), which gave the above result only for symmetric bodies and achieved a dependence of 2^{O(n)}(1+log^{5\/2}(1\/eps)\/eps^3)^n. \r\n\r\nFor our methods, we reduce the problem of volume estimation in K to counting lattice points in K subseteq R^n (via enumeration) for a specially constructed lattice L: a so-called thin covering of space with respect to K (more precisely, for which L + K = R^n and vol_n(K)\/det(L) = 2^{O(n)}). The trade off between time and approximation ratio is achieved by scaling down the lattice. \r\n\r\nAs our main technical contribution, we give the first deterministic 2^{O(n)}-time and poly(n)-space construction of thin covering lattices for general convex bodies. This improves on a recent construction of Alon et al (STOC 2013) which requires exponential space and only works for symmetric bodies. For our construction, we combine the use of the M-ellipsoid from convex geometry (Milman, C.R. Math. Acad. Sci. Paris 1986) together with lattice sparsification and densification techniques (Dadush and Kun, SODA 2013; Rogers, J. London Math. Soc. 1950).","keywords":["Deterministic Volume Estimation","Convex Geometry","Lattice Coverings of Space","Lattice Point Enumeration"],"author":{"@type":"Person","name":"Dadush, Daniel","givenName":"Daniel","familyName":"Dadush"},"position":53,"pageStart":704,"pageEnd":718,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Dadush, Daniel","givenName":"Daniel","familyName":"Dadush"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.704","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7899","name":"Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings","abstract":"We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).","keywords":["shallow cuttings","derandomization","halfspace range reporting","geometric data structures"],"author":[{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},{"@type":"Person","name":"Tsakalidis, Konstantinos","givenName":"Konstantinos","familyName":"Tsakalidis"}],"position":54,"pageStart":719,"pageEnd":732,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},{"@type":"Person","name":"Tsakalidis, Konstantinos","givenName":"Konstantinos","familyName":"Tsakalidis"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.719","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7900","name":"A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions","abstract":"Chazelle [FOCS'89] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same.","keywords":["convex polyhedra","intersection","Dobkin\u2013Kirkpatrick hierarchy"],"author":{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},"position":55,"pageStart":733,"pageEnd":738,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.733","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7901","name":"Approximability of the Discrete Fr\u00e9chet Distance","abstract":"The Fr\u00e9chet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.\r\n\r\nIn this paper, we study the approximability of the discrete Fr\u00e9chet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Fr\u00e9chet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.\r\n\r\nThis raises the question of how well we can approximate the Fr\u00e9chet distance (of two given d-dimensional point sequences of length n) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2^Theta(n). Moreover, we design an alpha-approximation algorithm that runs in time O(n log n + n^2 \/ alpha), for any alpha in [1, n]. Hence, an n^epsilon-approximation of the Fr\u00e9chet distance can be computed in strongly subquadratic time, for any epsilon > 0.","keywords":["Fr\u00e9chet distance","approximation","lower bounds","Strong Exponential Time Hypothesis"],"author":[{"@type":"Person","name":"Bringmann, Karl","givenName":"Karl","familyName":"Bringmann"},{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"}],"position":56,"pageStart":739,"pageEnd":753,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bringmann, Karl","givenName":"Karl","familyName":"Bringmann"},{"@type":"Person","name":"Mulzer, Wolfgang","givenName":"Wolfgang","familyName":"Mulzer"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.739","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7902","name":"The Hardness of Approximation of Euclidean k-Means","abstract":"The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k\/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d.\r\n\r\nIn this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.","keywords":["Euclidean k-means","Hardness of Approximation","Vertex Cover"],"author":[{"@type":"Person","name":"Awasthi, Pranjal","givenName":"Pranjal","familyName":"Awasthi"},{"@type":"Person","name":"Charikar, Moses","givenName":"Moses","familyName":"Charikar"},{"@type":"Person","name":"Krishnaswamy, Ravishankar","givenName":"Ravishankar","familyName":"Krishnaswamy"},{"@type":"Person","name":"Sinop, Ali Kemal","givenName":"Ali Kemal","familyName":"Sinop"}],"position":57,"pageStart":754,"pageEnd":767,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Awasthi, Pranjal","givenName":"Pranjal","familyName":"Awasthi"},{"@type":"Person","name":"Charikar, Moses","givenName":"Moses","familyName":"Charikar"},{"@type":"Person","name":"Krishnaswamy, Ravishankar","givenName":"Ravishankar","familyName":"Krishnaswamy"},{"@type":"Person","name":"Sinop, Ali Kemal","givenName":"Ali Kemal","familyName":"Sinop"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.754","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7903","name":"A Fire Fighter\u2019s Problem","abstract":"Suppose that a circular fire spreads in the plane at unit speed. A fire fighter can build a barrier at speed v > 1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We provide two results. First, we analyze the natural strategy where the fighter keeps building a barrier along the frontier of the expanding fire. We prove that this approach contains the fire if v > v_c = 2.6144... holds. Second, we show that any \"spiralling\" strategy must have speed v > 1.618, the golden ratio, in order to succeed.","keywords":["Motion Planning","Dynamic Environments","Spiralling strategies","Lower and upper bounds"],"author":[{"@type":"Person","name":"Klein, Rolf","givenName":"Rolf","familyName":"Klein"},{"@type":"Person","name":"Langetepe, Elmar","givenName":"Elmar","familyName":"Langetepe"},{"@type":"Person","name":"Levcopoulos, Christos","givenName":"Christos","familyName":"Levcopoulos"}],"position":58,"pageStart":768,"pageEnd":780,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Klein, Rolf","givenName":"Rolf","familyName":"Klein"},{"@type":"Person","name":"Langetepe, Elmar","givenName":"Elmar","familyName":"Langetepe"},{"@type":"Person","name":"Levcopoulos, Christos","givenName":"Christos","familyName":"Levcopoulos"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.768","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7904","name":"Approximate Geometric MST Range Queries","abstract":"Range searching is a widely-used method in computational geometry for efficiently accessing local regions of a large data set. Typically, range searching involves either counting or reporting the points lying within a given query region, but it is often desirable to compute statistics that better describe the structure of the point set lying within the region, not just the count.\r\n\r\nIn this paper we consider the geometric minimum spanning tree (MST) problem in the context of range searching where approximation is allowed. We are given a set P of n points in R^d. The objective is to preprocess P so that given an admissible query region Q, it is possible to efficiently approximate the weight of the minimum spanning tree of the subset of P lying within Q. There are two natural sources of approximation error, first by treating Q as a fuzzy object and second by approximating the MST weight itself. To model this, we assume that we are given two positive real approximation parameters eps_q and eps_w. Following the typical practice in approximate range searching, the range is expressed as two shapes Q^- and Q^+, where Q^- is contained in Q which is contained in Q^+, and their boundaries are separated by a distance of at least eps_q diam(Q). Points within Q^- must be included and points external to Q^+ cannot be included. A weight W is a valid answer to the query if there exist subsets P' and P'' of P, such that Q^- is contained in P' which is contained in P'' which is contained in Q^+ and wt(MST(P')) <= W <= (1+eps_w) wt(MST(P'')).\r\n\r\nIn this paper, we present an efficient data structure for answering such queries. Our approach uses simple data structures based on quadtrees, and it can be applied whenever Q^- and Q^+ are compact sets of constant combinatorial complexity. It uses space O(n), and it answers queries in time O(log n + 1\/(eps_q eps_w)^{d + O(1)}). The O(1) term is a small constant independent of dimension, and the hidden constant factor in the overall running time depends on d, but not on eps_q or eps_w. Preprocessing requires knowledge of eps_w, but not eps_q.","keywords":["Geometric data structures","Minimum spanning trees","Range searching","Approximation algorithms"],"author":[{"@type":"Person","name":"Arya, Sunil","givenName":"Sunil","familyName":"Arya"},{"@type":"Person","name":"Mount, David M.","givenName":"David M.","familyName":"Mount"},{"@type":"Person","name":"Park, Eunhui","givenName":"Eunhui","familyName":"Park"}],"position":59,"pageStart":781,"pageEnd":795,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arya, Sunil","givenName":"Sunil","familyName":"Arya"},{"@type":"Person","name":"Mount, David M.","givenName":"David M.","familyName":"Mount"},{"@type":"Person","name":"Park, Eunhui","givenName":"Eunhui","familyName":"Park"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.781","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7905","name":"Maintaining Contour Trees of Dynamic Terrains","abstract":"We study the problem of maintaining the contour tree T of a terrain Sigma, represented as a triangulated xy-monotone surface, as the heights of its vertices vary continuously with time. We characterize the combinatorial changes in T and how they relate to topological changes in Sigma. We present a kinetic data structure (KDS) for maintaining T efficiently. It maintains certificates that fail, i.e., an event occurs, only when the heights of two adjacent vertices become equal or two saddle vertices appear on the same contour. Assuming that the heights of two vertices of Sigma become equal only O(1) times and these instances can be computed in O(1) time, the KDS processes O(kappa + n) events, where n is the number of vertices in Sigma and kappa is the number of events at which the combinatorial structure of T changes, and processes each event in O(log n) time. The KDS can be extended to maintain an augmented contour tree and a join\/split tree.","keywords":["Contour tree","dynamic terrain","kinetic data structure"],"author":[{"@type":"Person","name":"Agarwal, Pankaj K.","givenName":"Pankaj K.","familyName":"Agarwal"},{"@type":"Person","name":"M\u00f8lhave, Thomas","givenName":"Thomas","familyName":"M\u00f8lhave"},{"@type":"Person","name":"Revsb\u00e6k, Morten","givenName":"Morten","familyName":"Revsb\u00e6k"},{"@type":"Person","name":"Safa, Issam","givenName":"Issam","familyName":"Safa"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"},{"@type":"Person","name":"Yang, Jungwoo","givenName":"Jungwoo","familyName":"Yang"}],"position":60,"pageStart":796,"pageEnd":811,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Agarwal, Pankaj K.","givenName":"Pankaj K.","familyName":"Agarwal"},{"@type":"Person","name":"M\u00f8lhave, Thomas","givenName":"Thomas","familyName":"M\u00f8lhave"},{"@type":"Person","name":"Revsb\u00e6k, Morten","givenName":"Morten","familyName":"Revsb\u00e6k"},{"@type":"Person","name":"Safa, Issam","givenName":"Issam","familyName":"Safa"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"},{"@type":"Person","name":"Yang, Jungwoo","givenName":"Jungwoo","familyName":"Yang"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.796","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7906","name":"Hyperorthogonal Well-Folded Hilbert Curves","abstract":"R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes - smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Omega(2^{d\/2}) larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.","keywords":["space-filling curve","Hilbert curve","multi-dimensional","range query","R-tree"],"author":[{"@type":"Person","name":"Bos, Arie","givenName":"Arie","familyName":"Bos"},{"@type":"Person","name":"Haverkort, Herman J.","givenName":"Herman J.","familyName":"Haverkort"}],"position":61,"pageStart":812,"pageEnd":826,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bos, Arie","givenName":"Arie","familyName":"Bos"},{"@type":"Person","name":"Haverkort, Herman J.","givenName":"Herman J.","familyName":"Haverkort"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.812","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/web.archive.org\/web\/www.caam.rice.edu\/~dougm\/twiddle\/Hilbert\/","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7907","name":"Topological Analysis of Scalar Fields with Outliers","abstract":"Given a real-valued function f defined over a manifold M embedded in R^d, we are interested in recovering structural information about f from the sole information of its values on a finite sample P. Existing methods provide approximation to the persistence diagram of f when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice.\r\n\r\nWe propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.","keywords":["Persistent Homology","Topological Data Analysis","Scalar Field Analysis","Nested Rips Filtration","Distance to a Measure"],"author":[{"@type":"Person","name":"Buchet, Micka\u00ebl","givenName":"Micka\u00ebl","familyName":"Buchet"},{"@type":"Person","name":"Chazal, Fr\u00e9d\u00e9ric","givenName":"Fr\u00e9d\u00e9ric","familyName":"Chazal"},{"@type":"Person","name":"Dey, Tamal K.","givenName":"Tamal K.","familyName":"Dey"},{"@type":"Person","name":"Fan, Fengtao","givenName":"Fengtao","familyName":"Fan"},{"@type":"Person","name":"Oudot, Steve Y.","givenName":"Steve Y.","familyName":"Oudot"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"position":62,"pageStart":827,"pageEnd":841,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Buchet, Micka\u00ebl","givenName":"Micka\u00ebl","familyName":"Buchet"},{"@type":"Person","name":"Chazal, Fr\u00e9d\u00e9ric","givenName":"Fr\u00e9d\u00e9ric","familyName":"Chazal"},{"@type":"Person","name":"Dey, Tamal K.","givenName":"Tamal K.","familyName":"Dey"},{"@type":"Person","name":"Fan, Fengtao","givenName":"Fengtao","familyName":"Fan"},{"@type":"Person","name":"Oudot, Steve Y.","givenName":"Steve Y.","familyName":"Oudot"},{"@type":"Person","name":"Wang, Yusu","givenName":"Yusu","familyName":"Wang"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.827","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7908","name":"On Computability and Triviality of Well Groups","abstract":"The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1.\r\n\r\nOur contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set.\r\n\r\nFor the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact.\r\n\r\nFor the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.","keywords":["nonlinear equations","robustness","well groups","computation","homotopy theory"],"author":[{"@type":"Person","name":"Franek, Peter","givenName":"Peter","familyName":"Franek"},{"@type":"Person","name":"Krc\u00e1l, Marek","givenName":"Marek","familyName":"Krc\u00e1l"}],"position":63,"pageStart":842,"pageEnd":856,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Franek, Peter","givenName":"Peter","familyName":"Franek"},{"@type":"Person","name":"Krc\u00e1l, Marek","givenName":"Marek","familyName":"Krc\u00e1l"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.842","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7909","name":"Geometric Inference on Kernel Density Estimates","abstract":"We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to superlevel sets of the kernel density estimate. We prove new properties about the kernel distance, demonstrating stability results and allowing it to inherit reconstruction results from recent advances in distance-based topological reconstruction. Moreover, we provide an algorithm to estimate its topology using weighted Vietoris-Rips complexes.","keywords":["topological data analysis","kernel density estimate","kernel distance"],"author":[{"@type":"Person","name":"Phillips, Jeff M.","givenName":"Jeff M.","familyName":"Phillips"},{"@type":"Person","name":"Wang, Bei","givenName":"Bei","familyName":"Wang"},{"@type":"Person","name":"Zheng, Yan","givenName":"Yan","familyName":"Zheng"}],"position":64,"pageStart":857,"pageEnd":871,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Phillips, Jeff M.","givenName":"Jeff M.","familyName":"Phillips"},{"@type":"Person","name":"Wang, Bei","givenName":"Bei","familyName":"Wang"},{"@type":"Person","name":"Zheng, Yan","givenName":"Yan","familyName":"Zheng"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.857","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"},{"@type":"ScholarlyArticle","@id":"#article7910","name":"Modeling Real-World Data Sets (Invited Talk)","abstract":"Traditionally, the performance of algorithms is evaluated using worst-case analysis. For a number of problems, this type of analysis gives overly pessimistic results: Worst-case inputs are rather artificial and do not occur in practical applications. In this lecture we review some alternative analysis approaches leading to more realistic and robust performance evaluations. \r\n\r\nSpecifically, we focus on the approach of modeling real-world data sets. We report on two studies performed by the author for the problems of self-organizing search and paging. In these settings real data sets exhibit locality of reference. We devise mathematical models capturing locality. Furthermore, we present combined theoretical and experimental analyses in which the theoretically proven and experimentally observed performance guarantees match up to very small relative errors.","keywords":["Worst-case analysis","real data sets","locality of reference","paging","self-organizing lists"],"author":{"@type":"Person","name":"Albers, Susanne","givenName":"Susanne","familyName":"Albers"},"position":65,"pageStart":872,"pageEnd":872,"dateCreated":"2015-06-12","datePublished":"2015-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Albers, Susanne","givenName":"Susanne","familyName":"Albers"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SOCG.2015.872","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6237"}]}