Martin Fink, J. Hershberger, Nirman Kumar, S. Suri
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.
我们描述了一个O(n^d)时间算法，用于计算两个d维概率点集线性可分的精确概率，对于任何固定的d >= 2。d空间中的概率点是通常的点，但具有相关的(独立的)存在概率。我们还证明了d维可分性问题等价于(d+1)维凸包隶属性问题，该问题要求查询点位于n个概率点的凸包内的概率。通过这种约简，我们将凸包隶属度的当前最佳界提高了n倍[Agarwal等人，ESA, 2014]。此外，我们的算法可以处理“输入退化”，其中超过k+1个点可能位于k维子空间，从而解决了[Agarwal et al.， ESA, 2014]中的一个开放问题。最后，我们通过k-SUM问题的简化证明了可分性问题的下界，这特别表明我们的二维可分性和三维凸壳隶属度的O(n^2)算法几乎是最优的。
{"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
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.
我们重新审视最小链接路径问题:给定一个多面体域和其中的两个点，用一条边数最少的多边形路径将这两个点连接起来。我们考虑路径的顶点和/或边缘被限制在域的边界上或可以在其内部的设置。我们的结果包括位复杂度界限，一个新的一般硬度结构，和一个多项式时间逼近方案。我们完全描述了二维的情况，并为问题的几个变体提供了三维和更高维度的第一个结果。具体地说，我们的结果解决了几个悬而未决的问题。我们证明了在计算机图形学中由光线追踪驱动的最小链路漫反射路径的计算是np困难的，即使对于带孔的二维多边形域也是如此。这仍然是一个悬而未决的问题[Ghosh et al.'2012]，尽管在这个主题上有大量的工作。我们还解决了手册[Goodman and Rourke'2004](见第27.5章，开放问题3)和开放问题项目[http://maven.smith.edu/~orourke/TOPP/](见问题22)中提到的[Mitchell et al.'1992]中的开放问题:“三维空间中最小链接路径问题的复杂性是多少?”我们的结果表明，即使在地形上，问题也是NP困难的(因此，由于答案的离散性，除非P=NP，否则不存在FPTAS)，但承认存在PTAS。
{"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}
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.
{"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}
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 $varepsilon > 0$.
Frechet距离是点序列和曲线的一种流行和广泛的距离度量。大约两年前，Agarwal等人[SIAM J. Comput. 2014]提出了一种新的(温和的)次二次算法来解决该问题的离散版本。这引发了一系列的活动，产生了几个新的算法和下界。本文研究了离散Frechet距离的近似性。基于Bringmann [FOCS 2014]最近的结果，我们提出了一个新的条件下界，表明离散Frechet距离的强次二次算法不太可能存在，即使在一维情况下，即使解可能近似到1.399的因子。这就提出了一个问题，我们如何在强次二次时间内很好地近似Frechet距离(两个给定的$d$ -维长度为$n$的点序列)。在此之前，没有已知的一般结果。我们通过分析一个简单的线性时间贪心算法的近似比为$2^{Theta(n)}$，提出了第一个这样的算法。此外，我们设计了一个$alpha$ -近似算法，该算法运行在时间$O(nlog n + n^2/alpha)$上，适用于任何$alphain [1, n]$。因此，对于任何$varepsilon > 0$, Frechet距离的$n^varepsilon$ -近似可以在强次二次时间内计算。
{"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
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.
{"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}
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.
{"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}
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. We 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.
{"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
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).
{"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
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.
{"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}
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.
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：