72 Search Results for "Barequet, Gill"


Volume

LIPIcs, Volume 129

35th International Symposium on Computational Geometry (SoCG 2019)

SoCG 2019, June 18-21, 2019, Portland, Oregon, USA

Editors: Gill Barequet and Yusu Wang

Document
Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

Authors: Gill Barequet, Evanthia Papadopoulou, and Martin Suderland

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)


Abstract
We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2).

Cite as

Gill Barequet, Evanthia Papadopoulou, and Martin Suderland. Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{barequet_et_al:LIPIcs.ISAAC.2019.62,
  author =	{Barequet, Gill and Papadopoulou, Evanthia and Suderland, Martin},
  title =	{{Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.62},
  URN =		{urn:nbn:de:0030-drops-115582},
  doi =		{10.4230/LIPIcs.ISAAC.2019.62},
  annote =	{Keywords: Voronoi diagram, lines, line segments, higher-order, order-k, unbounded, hypersphere arrangement, great hyperspheres}
}
Document
Complete Volume
LIPIcs, Volume 129, SoCG'19, Complete Volume

Authors: Gill Barequet and Yusu Wang

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
LIPIcs, Volume 129, SoCG'19, Complete Volume

Cite as

35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@Proceedings{barequet_et_al:LIPIcs.SoCG.2019,
  title =	{{LIPIcs, Volume 129, SoCG'19, Complete Volume}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019},
  URN =		{urn:nbn:de:0030-drops-105562},
  doi =		{10.4230/LIPIcs.SoCG.2019},
  annote =	{Keywords: Theory of computation, Computational geometry, Design and analysis of algorithms, Mathematics of computing, Combinatorics, Graph algortihms}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Gill Barequet and Yusu Wang

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
Front Matter, Table of Contents, Preface, Conference Organization

Cite as

35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{barequet_et_al:LIPIcs.SoCG.2019.0,
  author =	{Barequet, Gill and Wang, Yusu},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{0:i--0:xvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.0},
  URN =		{urn:nbn:de:0030-drops-104047},
  doi =		{10.4230/LIPIcs.SoCG.2019.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
A Geometric Data Structure from Neuroscience (Invited Talk)

Authors: Sanjoy Dasgupta

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
An intriguing geometric primitive, "expand-and-sparsify", has been found in the olfactory system of the fly and several other organisms. It maps an input vector to a much higher-dimensional sparse representation, using a random linear transformation followed by winner-take-all thresholding. I will show that this representation has a variety of formal properties, such as locality preservation, that make it an attractive data structure for algorithms and machine learning. In particular, mimicking the fly’s circuitry yields algorithms for similarity search and for novelty detection that have provable guarantees as well as having practical performance that is competitive with state-of-the-art methods. This talk is based on work with Saket Navlakha (Salk Institute), Chuck Stevens (Salk Institute), and Chris Tosh (Columbia).

Cite as

Sanjoy Dasgupta. A Geometric Data Structure from Neuroscience (Invited Talk). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{dasgupta:LIPIcs.SoCG.2019.1,
  author =	{Dasgupta, Sanjoy},
  title =	{{A Geometric Data Structure from Neuroscience}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.1},
  URN =		{urn:nbn:de:0030-drops-104055},
  doi =		{10.4230/LIPIcs.SoCG.2019.1},
  annote =	{Keywords: Geometric data structure, algorithm design, neuroscience}
}
Document
Invited Talk
Some Geometric and Computational Challenges Arising in Structural Molecular Biology (Invited Talk)

Authors: Bruce R. Donald

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous geometry to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials, one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited geometric information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions. We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. For more information please see Y. Qi et al. Jour. Mol. Biol. 2018; 430(18 Pt B):3412-3426. doi: 10.1016/j.jmb.2018.06.022. Ultimately, this method has parallels to other branches of geometric computing that balance discrete and continuous representations, including physical geometric algorithms, robotics, computational geometry, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

Cite as

Bruce R. Donald. Some Geometric and Computational Challenges Arising in Structural Molecular Biology (Invited Talk). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{donald:LIPIcs.SoCG.2019.2,
  author =	{Donald, Bruce R.},
  title =	{{Some Geometric and Computational Challenges Arising in Structural Molecular Biology}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.2},
  URN =		{urn:nbn:de:0030-drops-104065},
  doi =		{10.4230/LIPIcs.SoCG.2019.2},
  annote =	{Keywords: Geometric computing, combutational biology, Continuous Interdomain Orientation Distributions of Protein Conformations}
}
Document
A New Lower Bound for Semigroup Orthogonal Range Searching

Authors: Peyman Afshani

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [Yao, 1982]: using m units of space, the query time Q(n) must be Omega(alpha(m,n) + n/(m-n+1)) where alpha(*,*) is the inverse Ackermann’s function, a very slowly growing function. In d dimensions, Bernard Chazelle [Chazelle, 1990] proved that the query time must be Q(n) = Omega((log_beta n)^{d-1}) where beta = 2m/n. Chazelle’s lower bound is known to be tight for when space consumption is "high" i.e., m = Omega(n log^{d+epsilon}n). We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for "low space": we prove that we must have m Q(n) = Omega(n (log n log log n)^{d-1}). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.

Cite as

Peyman Afshani. A New Lower Bound for Semigroup Orthogonal Range Searching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{afshani:LIPIcs.SoCG.2019.3,
  author =	{Afshani, Peyman},
  title =	{{A New Lower Bound for Semigroup Orthogonal Range Searching}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.3},
  URN =		{urn:nbn:de:0030-drops-104075},
  doi =		{10.4230/LIPIcs.SoCG.2019.3},
  annote =	{Keywords: Data Structures, Range Searching, Lower bounds}
}
Document
Independent Range Sampling, Revisited Again

Authors: Peyman Afshani and Jeff M. Phillips

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer k, we can extract k independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight. This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries. We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to n^{2/3}, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.

Cite as

Peyman Afshani and Jeff M. Phillips. Independent Range Sampling, Revisited Again. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{afshani_et_al:LIPIcs.SoCG.2019.4,
  author =	{Afshani, Peyman and Phillips, Jeff M.},
  title =	{{Independent Range Sampling, Revisited Again}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.4},
  URN =		{urn:nbn:de:0030-drops-104088},
  doi =		{10.4230/LIPIcs.SoCG.2019.4},
  annote =	{Keywords: Range Searching, Data Structures, Sampling}
}
Document
An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

Authors: Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in R^d and if D >= 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of R^d \ Z(P) intersects O(n/D^{d-g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently - the expected running time of our algorithm is linear in |S|. Our approach exploits the technique of quantifier elimination combined with that of epsilon-samples. We present four applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in R^d in O(log n) time, with storage complexity and expected preprocessing time of O(n^{d+epsilon}). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(n^{t+epsilon}) storage and expected preprocessing time, where t > 0 is an integer that depends on d and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R^{d} in O(log^2 n) time, with O(n^{d+epsilon}) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

Cite as

Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.5,
  author =	{Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Zahl, Joshua},
  title =	{{An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.5},
  URN =		{urn:nbn:de:0030-drops-104096},
  doi =		{10.4230/LIPIcs.SoCG.2019.5},
  annote =	{Keywords: Polynomial partitioning, quantifier elimination, semi-algebraic range spaces, epsilon-samples}
}
Document
Efficient Algorithms for Geometric Partial Matching

Authors: Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
Let A and B be two point sets in the plane of sizes r and n respectively (assume r <= n), and let k be a parameter. A matching between A and B is a family of pairs in A x B so that any point of A cup B appears in at most one pair. Given two positive integers p and q, we define the cost of matching M to be c(M) = sum_{(a, b) in M}||a-b||_p^q where ||*||_p is the L_p-norm. The geometric partial matching problem asks to find the minimum-cost size-k matching between A and B. We present efficient algorithms for geometric partial matching problem that work for any powers of L_p-norm matching objective: An exact algorithm that runs in O((n + k^2)polylog n) time, and a (1 + epsilon)-approximation algorithm that runs in O((n + k sqrt{k})polylog n * log epsilon^{-1}) time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in O(min{n^2, rn^{3/2}}polylog n) time.

Cite as

Pankaj K. Agarwal, Hsien-Chih Chang, and Allen Xiao. Efficient Algorithms for Geometric Partial Matching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2019.6,
  author =	{Agarwal, Pankaj K. and Chang, Hsien-Chih and Xiao, Allen},
  title =	{{Efficient Algorithms for Geometric Partial Matching}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.6},
  URN =		{urn:nbn:de:0030-drops-104109},
  doi =		{10.4230/LIPIcs.SoCG.2019.6},
  annote =	{Keywords: partial matching, transportation, optimal transport, minimum-cost flow, bichromatic closest pair}
}
Document
Connecting the Dots (with Minimum Crossings)

Authors: Akanksha Agrawal, Grzegorz Guśpiel, Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
We study a prototype Crossing Minimization problem, defined as follows. Let F be an infinite family of (possibly vertex-labeled) graphs. Then, given a set P of (possibly labeled) n points in the Euclidean plane, a collection L subseteq Lines(P)={l: l is a line segment with both endpoints in P}, and a non-negative integer k, decide if there is a subcollection L'subseteq L such that the graph G=(P,L') is isomorphic to a graph in F and L' has at most k crossings. By G=(P,L'), we refer to the graph on vertex set P, where two vertices are adjacent if and only if there is a line segment that connects them in L'. Intuitively, in Crossing Minimization, we have a set of locations of interest, and we want to build/draw/exhibit connections between them (where L indicates where it is feasible to have these connections) so that we obtain a structure in F. Natural choices for F are the collections of perfect matchings, Hamiltonian paths, and graphs that contain an (s,t)-path (a path whose endpoints are labeled). While the objective of seeking a solution with few crossings is of interest from a theoretical point of view, it is also well motivated by a wide range of practical considerations. For example, links/roads (such as highways) may be cheaper to build and faster to traverse, and signals/moving objects would collide/interrupt each other less often. Further, graphs with fewer crossings are preferred for graphic user interfaces. As a starting point for a systematic study, we consider a special case of Crossing Minimization. Already for this case, we obtain NP-hardness and W[1]-hardness results, and ETH-based lower bounds. Specifically, suppose that the input also contains a collection D of d non-crossing line segments such that each point in P belongs to exactly one line in D, and L does not contain line segments between points on the same line in D. Clearly, Crossing Minimization is the case where d=n - then, P is in general position. The case of d=2 is of interest not only because it is the most restricted non-trivial case, but also since it corresponds to a class of graphs that has been well studied - specifically, it is Crossing Minimization where G=(P,L) is a (bipartite) graph with a so called two-layer drawing. For d=2, we consider three basic choices of F. For perfect matchings, we show (i) NP-hardness with an ETH-based lower bound, (ii) solvability in subexponential parameterized time, and (iii) existence of an O(k^2)-vertex kernel. Second, for Hamiltonian paths, we show (i) solvability in subexponential parameterized time, and (ii) existence of an O(k^2)-vertex kernel. Lastly, for graphs that contain an (s,t)-path, we show (i) NP-hardness and W[1]-hardness, and (ii) membership in XP.

Cite as

Akanksha Agrawal, Grzegorz Guśpiel, Jayakrishnan Madathil, Saket Saurabh, and Meirav Zehavi. Connecting the Dots (with Minimum Crossings). In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{agrawal_et_al:LIPIcs.SoCG.2019.7,
  author =	{Agrawal, Akanksha and Gu\'{s}piel, Grzegorz and Madathil, Jayakrishnan and Saurabh, Saket and Zehavi, Meirav},
  title =	{{Connecting the Dots (with Minimum Crossings)}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.7},
  URN =		{urn:nbn:de:0030-drops-104117},
  doi =		{10.4230/LIPIcs.SoCG.2019.7},
  annote =	{Keywords: crossing minimization, parameterized complexity, FPT algorithm, polynomial kernel, W\lbrack1\rbrack-hardness}
}
Document
General Techniques for Approximate Incidences and Their Application to the Camera Posing Problem

Authors: Dror Aiger, Haim Kaplan, Efi Kokiopoulou, Micha Sharir, and Bernhard Zeisl

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
We consider the classical camera pose estimation problem that arises in many computer vision applications, in which we are given n 2D-3D correspondences between points in the scene and points in the camera image (some of which are incorrect associations), and where we aim to determine the camera pose (the position and orientation of the camera in the scene) from this data. We demonstrate that this posing problem can be reduced to the problem of computing epsilon-approximate incidences between two-dimensional surfaces (derived from the input correspondences) and points (on a grid) in a four-dimensional pose space. Similar reductions can be applied to other camera pose problems, as well as to similar problems in related application areas. We describe and analyze three techniques for solving the resulting epsilon-approximate incidences problem in the context of our camera posing application. The first is a straightforward assignment of surfaces to the cells of a grid (of side-length epsilon) that they intersect. The second is a variant of a primal-dual technique, recently introduced by a subset of the authors [Aiger et al., 2017] for different (and simpler) applications. The third is a non-trivial generalization of a data structure Fonseca and Mount [Da Fonseca and Mount, 2010], originally designed for the case of hyperplanes. We present and analyze this technique in full generality, and then apply it to the camera posing problem at hand. We compare our methods experimentally on real and synthetic data. Our experiments show that for the typical values of n and epsilon, the primal-dual method is the fastest, also in practice.

Cite as

Dror Aiger, Haim Kaplan, Efi Kokiopoulou, Micha Sharir, and Bernhard Zeisl. General Techniques for Approximate Incidences and Their Application to the Camera Posing Problem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{aiger_et_al:LIPIcs.SoCG.2019.8,
  author =	{Aiger, Dror and Kaplan, Haim and Kokiopoulou, Efi and Sharir, Micha and Zeisl, Bernhard},
  title =	{{General Techniques for Approximate Incidences and Their Application to the Camera Posing Problem}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.8},
  URN =		{urn:nbn:de:0030-drops-104129},
  doi =		{10.4230/LIPIcs.SoCG.2019.8},
  annote =	{Keywords: Camera positioning, Approximate incidences, Incidences}
}
Document
Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Authors: Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

Cite as

Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth. Circumscribing Polygons and Polygonizations for Disjoint Line Segments. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{akitaya_et_al:LIPIcs.SoCG.2019.9,
  author =	{Akitaya, Hugo A. and Korman, Matias and Rudoy, Mikhail and Souvaine, Diane L. and T\'{o}th, Csaba D.},
  title =	{{Circumscribing Polygons and Polygonizations for Disjoint Line Segments}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.9},
  URN =		{urn:nbn:de:0030-drops-104136},
  doi =		{10.4230/LIPIcs.SoCG.2019.9},
  annote =	{Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics}
}
Document
Morphing Contact Representations of Graphs

Authors: Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, and Vincenzo Roselli

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set.

Cite as

Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, and Vincenzo Roselli. Morphing Contact Representations of Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{angelini_et_al:LIPIcs.SoCG.2019.10,
  author =	{Angelini, Patrizio and Chaplick, Steven and Cornelsen, Sabine and Da Lozzo, Giordano and Roselli, Vincenzo},
  title =	{{Morphing Contact Representations of Graphs}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.10},
  URN =		{urn:nbn:de:0030-drops-104145},
  doi =		{10.4230/LIPIcs.SoCG.2019.10},
  annote =	{Keywords: Contact representations, Triangulations, Planar morphs, Schnyder woods}
}
Document
When Convexity Helps Collapsing Complexes

Authors: Dominique Attali, André Lieutier, and David Salinas

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)


Abstract
This paper illustrates how convexity hypotheses help collapsing simplicial complexes. We first consider a collection of compact convex sets and show that the nerve of the collection is collapsible whenever the union of sets in the collection is convex. We apply this result to prove that the Delaunay complex of a finite point set is collapsible. We then consider a convex domain defined as the convex hull of a finite point set. We show that if the point set samples sufficiently densely the domain, then both the Cech complex and the Rips complex of the point set are collapsible for a well-chosen scale parameter. A key ingredient in our proofs consists in building a filtration by sweeping space with a growing sphere whose center has been fixed and studying events occurring through the filtration. Since the filtration mimics the sublevel sets of a Morse function with a single critical point, we anticipate this work to lay the foundations for a non-smooth, discrete Morse Theory.

Cite as

Dominique Attali, André Lieutier, and David Salinas. When Convexity Helps Collapsing Complexes. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Copy BibTex To Clipboard

@InProceedings{attali_et_al:LIPIcs.SoCG.2019.11,
  author =	{Attali, Dominique and Lieutier, Andr\'{e} and Salinas, David},
  title =	{{When Convexity Helps Collapsing Complexes}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.11},
  URN =		{urn:nbn:de:0030-drops-104152},
  doi =		{10.4230/LIPIcs.SoCG.2019.11},
  annote =	{Keywords: collapsibility, convexity, collection of compact convex sets, nerve, filtration, Delaunay complex, Cech complex, Rips complex}
}
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