Martin Fink, J. Hershberger, Nirman Kumar, S. Suri
{"title":"Hyperplane separability and convexity of probabilistic point sets","authors":"Martin Fink, J. Hershberger, Nirman Kumar, S. Suri","doi":"10.20382/jocg.v8i2a3","DOIUrl":"https://doi.org/10.20382/jocg.v8i2a3","url":null,"abstract":"We describe an O(n^d) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d+1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [Agarwal et al., ESA, 2014]. In addition, our algorithms can handle \"input degeneracies\" in which more than k+1 points may lie on a k-dimensional subspace, thus resolving an open problem in [Agarwal et al., ESA, 2014]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n^2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"25 1","pages":"38:1-38:16"},"PeriodicalIF":0.3,"publicationDate":"2016-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75050292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Kostitsyna, M. Löffler, V. Polishchuk, F. Staals
{"title":"On the complexity of minimum-link path problems","authors":"I. Kostitsyna, M. Löffler, V. Polishchuk, F. Staals","doi":"10.20382/jocg.v8i2a5","DOIUrl":"https://doi.org/10.20382/jocg.v8i2a5","url":null,"abstract":"We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2 dimensions, and provide first results in dimensions 3 and higher for several variants of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al.'2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al.'1992] mentioned in the handbook [Goodman and Rourke'2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [http://maven.smith.edu/~orourke/TOPP/] (see Problem 22): \"What is the complexity of the minimum-link path problem in 3-space?\" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"113 1","pages":"49:1-49:16"},"PeriodicalIF":0.3,"publicationDate":"2016-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79842845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hyperorthogonal well-folded Hilbert curves","authors":"A. Bos, H. Haverkort","doi":"10.20382/jocg.v7i2a7","DOIUrl":"https://doi.org/10.20382/jocg.v7i2a7","url":null,"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.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"1 1","pages":"812-826"},"PeriodicalIF":0.3,"publicationDate":"2015-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75952456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximability of the discrete Fréchet distance","authors":"K. Bringmann, Wolfgang Mulzer","doi":"10.20382/jocg.v7i2a4","DOIUrl":"https://doi.org/10.20382/jocg.v7i2a4","url":null,"abstract":"The Frechet 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. In this paper, we study the approximability of the discrete Frechet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Frechet 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. This raises the question of how well we can approximate the Frechet 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 $O(nlog n + n^2/alpha)$, for any $alphain [1, n]$. Hence, an $n^varepsilon$-approximation of the Frechet distance can be computed in strongly subquadratic time, for any $varepsilon > 0$.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"88 2","pages":"739-753"},"PeriodicalIF":0.3,"publicationDate":"2015-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72434929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Polishchuk, E. Arkin, A. Efrat, Christian Knauer, Joseph B. M. Mitchell, G. Rote, Lena Schlipf, Topi Talvitie
{"title":"Shortest path to a segment and quickest visibility queries","authors":"V. Polishchuk, E. Arkin, A. Efrat, Christian Knauer, Joseph B. M. Mitchell, G. Rote, Lena Schlipf, Topi Talvitie","doi":"10.20382/jocg.v7i2a5","DOIUrl":"https://doi.org/10.20382/jocg.v7i2a5","url":null,"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.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"6 1","pages":"658-673"},"PeriodicalIF":0.3,"publicationDate":"2015-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72710586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Recognizing shrinkable complexes is NP-complete","authors":"D. Attali, O. Devillers, M. Glisse, S. Lazard","doi":"10.20382/jocg.v7i1a18","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a18","url":null,"abstract":"We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (three-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes which are not shrinkable.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"657 1","pages":"74-86"},"PeriodicalIF":0.3,"publicationDate":"2014-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74734621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On self-approaching and increasing-chord drawings of 3-connected planar graphs","authors":"M. Nöllenburg, Roman Prutkin, Ignaz Rutter","doi":"10.20382/jocg.v7i1a3","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a3","url":null,"abstract":"An st-path in a drawing of a graph is self-approaching if during a traversal of the corresponding curve from s to any point t' on the curve the distance to t' is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching increasing-chord if any pair of vertices is connected by a self-approaching increasing-chord path. \u0000 \u0000We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. Moreover, we give a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"202 1","pages":"476-487"},"PeriodicalIF":0.3,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77009916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Eppstein, Danny Holten, M. Löffler, M. Nöllenburg, B. Speckmann, Kevin Verbeek
{"title":"Strict confluent drawing","authors":"D. Eppstein, Danny Holten, M. Löffler, M. Nöllenburg, B. Speckmann, Kevin Verbeek","doi":"10.20382/jocg.v7i1a2","DOIUrl":"https://doi.org/10.20382/jocg.v7i1a2","url":null,"abstract":"We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"5 1","pages":"352-363"},"PeriodicalIF":0.3,"publicationDate":"2013-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72513312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
K. Buchin, M. Buchin, M. V. Kreveld, B. Speckmann, F. Staals
{"title":"Trajectory grouping structure","authors":"K. Buchin, M. Buchin, M. V. Kreveld, B. Speckmann, F. Staals","doi":"10.20382/jocg.v6i1a3","DOIUrl":"https://doi.org/10.20382/jocg.v6i1a3","url":null,"abstract":"The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"2008 1","pages":"219-230"},"PeriodicalIF":0.3,"publicationDate":"2013-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86234158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a conjecture of Lovász on circle-representations of simple 4-regular planar graphs","authors":"M. Bekos, Chrysanthi N. Raftopoulou","doi":"10.20382/jocg.v6i1a1","DOIUrl":"https://doi.org/10.20382/jocg.v6i1a1","url":null,"abstract":"Lovasz conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, (a) we affirmatively answer Lovasz's conjecture, if G is 3-connected, and, (b) we demonstrate an infinite class of connected 4-regular planar graphs which are not 3-connected and do not admit a realization as a system of circles.","PeriodicalId":43044,"journal":{"name":"Journal of Computational Geometry","volume":"9 2-4 1","pages":"138-149"},"PeriodicalIF":0.3,"publicationDate":"2012-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78398447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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