R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^{-1}(I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero.","keywords":["Mapper","Reeb Graph","Extended Persistence","Topological Data Analysis"],"author":[{"@type":"Person","name":"Carri\u00e8re, Mathieu","givenName":"Mathieu","familyName":"Carri\u00e8re"},{"@type":"Person","name":"Oudot, Steve","givenName":"Steve","familyName":"Oudot"}],"position":25,"pageStart":"25:1","pageEnd":"25:16","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Carri\u00e8re, Mathieu","givenName":"Mathieu","familyName":"Carri\u00e8re"},{"@type":"Person","name":"Oudot, Steve","givenName":"Steve","familyName":"Oudot"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.25","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8499","name":"Max-Sum Diversity Via Convex Programming","abstract":"Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function f that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) which is the sum of the pairwise distances in S, is in this context a prominent diversification measure. The corresponding diversity maximization is the \"max-sum\" or \"sum-sum\" diversification. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint.\r\n\r\nIn this paper, we present a PTAS for the max-sum diversity problem under a matroid constraint for distances d(.,.) of negative type. Distances of negative type are, for example, metric distances stemming from the l_2 and l_1 norms, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.\r\n\r\nOur algorithm is based on techniques developed in geometric algorithms like metric embeddings and convex optimization. We show that one can compute a fractional solution of the usually non-convex relaxation of the problem which yields an upper bound on the optimum integer solution. Starting from this fractional solution, we employ a deterministic rounding approach which only incurs a small loss in terms of objective, thus leading to a PTAS. This technique can be applied to other previously studied variants of the max-sum dispersion function, including combinations of diversity with linear-score maximization, improving over the previous constant-factor approximation algorithms.","keywords":["Geometric Dispersion","Embeddings","Approximation Algorithms","Convex Programming","Matroids"],"author":[{"@type":"Person","name":"Cevallos, Alfonso","givenName":"Alfonso","familyName":"Cevallos"},{"@type":"Person","name":"Eisenbrand, Friedrich","givenName":"Friedrich","familyName":"Eisenbrand"},{"@type":"Person","name":"Zenklusen, Rico","givenName":"Rico","familyName":"Zenklusen"}],"position":26,"pageStart":"26:1","pageEnd":"26:14","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Cevallos, Alfonso","givenName":"Alfonso","familyName":"Cevallos"},{"@type":"Person","name":"Eisenbrand, Friedrich","givenName":"Friedrich","familyName":"Eisenbrand"},{"@type":"Person","name":"Zenklusen, Rico","givenName":"Rico","familyName":"Zenklusen"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.26","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"https:\/\/kam.mff.cuni.cz\/~matousek\/ba-a4.pdf","isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8500","name":"Dynamic Streaming Algorithms for Epsilon-Kernels","abstract":"Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model).\r\n\r\nAndoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a \"polynomial method\", does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.","keywords":["coresets","streaming algorithms","dynamic algorithms","polynomial method","randomization","outliers"],"author":{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},"position":27,"pageStart":"27:1","pageEnd":"27:11","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.27","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1007\/s00454-007-9013-2","http:\/\/dx.doi.org\/10.1007\/978-3-642-15775-2_42","http:\/\/dx.doi.org\/10.1145\/2582112.2582161","http:\/\/dx.doi.org\/10.1145\/235809.235813","http:\/\/dx.doi.org\/10.1016\/j.comgeo.2005.10.002","http:\/\/dx.doi.org\/10.1007\/s00454-009-9165-3","http:\/\/dx.doi.org\/10.1142\/S0218195908002520","http:\/\/dx.doi.org\/10.1145\/1007352.1007413","http:\/\/dx.doi.org\/10.1145\/2591796.2591812","http:\/\/dx.doi.org\/10.1007\/s00453-010-9392-2"],"isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8501","name":"Two Approaches to Building Time-Windowed Geometric Data Structures","abstract":"Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively.\r\n\r\nOur first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.","keywords":["time window","geometric data structures","range searching","dynamic convex hull"],"author":[{"@type":"Person","name":"Chan, Timothy M.","givenName":"Timothy M.","familyName":"Chan"},{"@type":"Person","name":"Pratt, Simon","givenName":"Simon","familyName":"Pratt"}],"position":28,"pageStart":"28:1","pageEnd":"28:15","dateCreated":"2016-06-10","datePublished":"2016-06-10","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":"Pratt, Simon","givenName":"Simon","familyName":"Pratt"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.28","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1016\/0925-7721(91)90001-U","http:\/\/dx.doi.org\/10.1109\/SFCS.1994.365724","http:\/\/dx.doi.org\/10.1145\/363647.363652","http:\/\/dx.doi.org\/10.1145\/1706591.1706596","http:\/\/dx.doi.org\/10.1007\/BF01840440","http:\/\/dx.doi.org\/10.1007\/BF01840441","http:\/\/dx.doi.org\/10.1007\/BF02187740","http:\/\/dx.doi.org\/10.1007\/BF02579170","http:\/\/dx.doi.org\/10.1145\/359131.359132","http:\/\/dx.doi.org\/10.1137\/0401038"],"isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8502","name":"Untangling Planar Curves","abstract":"Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3\/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3\/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.","keywords":["computational topology","homotopy","planar graphs","Delta-Y transformations","defect","Reidemeister moves","tangles"],"author":[{"@type":"Person","name":"Chang, Hsien-Chih","givenName":"Hsien-Chih","familyName":"Chang"},{"@type":"Person","name":"Erickson, Jeff","givenName":"Jeff","familyName":"Erickson"}],"position":29,"pageStart":"29:1","pageEnd":"29:16","dateCreated":"2016-06-10","datePublished":"2016-06-10","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":"Erickson, Jeff","givenName":"Jeff","familyName":"Erickson"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.29","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1510.00571","http:\/\/arxiv.org\/abs\/1411.3308"],"isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8503","name":"Inserting Multiple Edges into a Planar Graph","abstract":"Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+F. \r\n\r\nFinding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F|=1 [Gutwenger et al, SODA 2001\/Algorithmica] and polynomial time solvable when all of F are incident to a new vertex [Chimani et al, SODA 2009]. The complexity for general F but with constant k=|F| was open, but algorithms both with relative and absolute approximation guarantees have been presented [Chuzhoy et al, SODA 2011], [Chimani-Hlineny, ICALP 2011]. We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V(G)|) time for any constant k.","keywords":["crossing number","edge insertion","parameterized complexity","path homotopy","funnel algorithm"],"author":[{"@type":"Person","name":"Chimani, Markus","givenName":"Markus","familyName":"Chimani"},{"@type":"Person","name":"Hlinen\u00fd, Petr","givenName":"Petr","familyName":"Hlinen\u00fd"}],"position":30,"pageStart":"30:1","pageEnd":"30:15","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Chimani, Markus","givenName":"Markus","familyName":"Chimani"},{"@type":"Person","name":"Hlinen\u00fd, Petr","givenName":"Petr","familyName":"Hlinen\u00fd"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.30","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/hdl.handle.net\/2003\/25955","isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8504","name":"Polynomial-Sized Topological Approximations Using the Permutahedron","abstract":"Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry. \r\n\r\nBuilding on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1\/(log^{1+c}n) for c in (0,1).","keywords":["Persistent Homology","Topological Data Analysis","Simplicial Approximation","Permutahedron","Approximation Algorithms"],"author":[{"@type":"Person","name":"Choudhary, Aruni","givenName":"Aruni","familyName":"Choudhary"},{"@type":"Person","name":"Kerber, Michael","givenName":"Michael","familyName":"Kerber"},{"@type":"Person","name":"Raghvendra, Sharath","givenName":"Sharath","familyName":"Raghvendra"}],"position":31,"pageStart":"31:1","pageEnd":"31:16","dateCreated":"2016-06-10","datePublished":"2016-06-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Choudhary, Aruni","givenName":"Aruni","familyName":"Choudhary"},{"@type":"Person","name":"Kerber, Michael","givenName":"Michael","familyName":"Kerber"},{"@type":"Person","name":"Raghvendra, Sharath","givenName":"Sharath","familyName":"Raghvendra"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2016.31","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/graphics.stanford.edu\/papers\/permutohedral\/permutohedral_techreport.pdf","http:\/\/dx.doi.org\/10.1007\/s00200-014-0247-y","http:\/\/dx.doi.org\/10.1007\/BF02776078","http:\/\/dx.doi.org\/10.1007\/s00454-014-9573-x","http:\/\/dx.doi.org\/10.1145\/1542362.1542407","http:\/\/arxiv.org\/abs\/1601.02732","http:\/\/people.mpi-inf.mpg.de\/~achoudha\/Files\/AppCech.pdf","http:\/\/dx.doi.org\/10.1145\/2582112.2582165","http:\/\/www.ams.org\/bookstore-getitem\/item=MBK-69","http:\/\/dx.doi.org\/10.1145\/77635.77639","http:\/\/dx.doi.org\/10.1007\/BF02764938","http:\/\/dx.doi.org\/10.1007\/s00454-013-9513-1"],"isPartOf":"#volume6254"},{"@type":"ScholarlyArticle","@id":"#article8505","name":"Faster Algorithms for Computing Plurality Points","abstract":"Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p' in R^d by a voter v in V if dist(v,p) < dist(v,p'). A point p is called a plurality point if it is preferred by at least as many voters as any other point p'.\r\n\r\nWe present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball.\r\n\r\nFinally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector