{"title":"On Reverse Shortest Paths in Geometric Proximity Graphs","authors":"Pankaj Agarwal, M. J. Katz, M. Sharir","doi":"10.4230/LIPIcs.ISAAC.2022.42","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2022.42","url":null,"abstract":"Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in R 2 , and let ϱ : S × S → R ≥ 0 be a distance function on S . For a parameter r ≥ 0, we define the proximity graph G ( r ) = ( S, E ) where E = { ( e 1 , e 2 ) ∈ S × S | e 1 ̸ = e 2 , ϱ ( e 1 , e 2 ) ≤ r } . Given S , s, t ∈ S , and an integer k ≥ 1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r ∗ ≥ 0 such that G ( r ∗ ) contains a path from s to t of length at most k . In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value r ≥ 0, determine whether G ( r ) contains a path from s to t of length at most k . Next, we adapt our decision algorithm and combine it with a random-sampling method to compute r ∗ , by efficiently performing a binary search over an implicit set of O ( n 2 ) candidate values that contains r ∗ . We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an O ∗ ( n 4 / 3 ) expected-time randomized algorithm (where O ∗ ( · ) hides polylog( n ) factors) for the case where S is a set of pairwise-disjoint line segments in R 2 and ϱ ( e 1 , e 2 ) = min x ∈ e 1 ,y ∈ e 2 ∥ x − y ∥ (where ∥ · ∥ is the Euclidean distance), and (ii","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"105 1","pages":"102053"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79260595","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":"Algorithms for Radius-Optimally Augmenting Trees in a Metric Space","authors":"Joachim Gudmundsson, Y. Sha","doi":"10.1007/978-3-030-83508-8_33","DOIUrl":"https://doi.org/10.1007/978-3-030-83508-8_33","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"32 1","pages":"102018"},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80978291","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":"Augmenting Graphs to Minimize the Radius","authors":"Joachim Gudmundsson, Y. Sha, Fan Yao","doi":"10.4230/LIPIcs.ISAAC.2021.45","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.45","url":null,"abstract":"We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time (5 / 3 − ϵ )-approximation algorithm, for any ϵ > 0, unless P = NP . We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"85 1","pages":"101996"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84074992","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":"Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces","authors":"K. Cho, Eunjin Oh","doi":"10.4230/LIPIcs.ISAAC.2021.46","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.46","url":null,"abstract":"In this paper, we present a linear-time approximation scheme for k -means clustering of incomplete data points in d -dimensional Euclidean space. An incomplete data point with ∆ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension ∆. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k -means clustering of axis-parallel affine subspaces of dimension ∆ that yields an (1+ ϵ )-approximate solution in O ( nd ) time. The constants hidden behind O ( · ) depend only on ∆ , ϵ and k . This improves the O ( n 2 d )-time algorithm by Eiben et al. [SODA’21] by a factor of n .","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"1 1","pages":"101981"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74431897","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":"Intersecting Disks Using Two Congruent Disks","authors":"Byeonguk Kang, J. Choi, Hee-Kap Ahn","doi":"10.1007/978-3-030-79987-8_28","DOIUrl":"https://doi.org/10.1007/978-3-030-79987-8_28","url":null,"abstract":"","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"40 1","pages":"101966"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86546354","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 path-greedy geometric spanners","authors":"W. Evans, Lucca Siaudzionis","doi":"10.14288/1.0402167","DOIUrl":"https://doi.org/10.14288/1.0402167","url":null,"abstract":"A t-spanner is a graph in which the shortest path between two vertices never exceeds t times the distance between the two nodes – a t-approximation of the complete graph. A geometric graph is one in which its vertices are points with defined coordinates and the edges correspond to line segments between them with a distance function, such as Euclidean distance. Geometric spanners are used to design networks of reduced complexity, optimizing metrics such as the planarity or degree of the graph. One famous algorithm used to generate spanners is path-greedy, which scans pairs of points in non-decreasing order of distance and adds the edge between them unless the current set of added edges already connects them with a path that tapproximates the edge length. Graphs from this algorithm are called path-greedy spanners. This work analyzes properties of path-greedy geometric spanners under different conditions. Specifically, we answer an open problem regarding the planarity and degree of path-greedy 5.19-spanners in convex point sets, and explore how the algorithm behaves under random tiebreaks for grid point sets. Lastly, we show a simple and efficient way to reduce the degree of a plane spanner by adding extra points.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"93 1","pages":"101948"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84227170","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}
Daniel Bertschinger, Meghana M. Reddy, Enrico Mann
{"title":"Lions and Contamination: Monotone Clearings","authors":"Daniel Bertschinger, Meghana M. Reddy, Enrico Mann","doi":"10.4230/LIPIcs.SWAT.2022.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.17","url":null,"abstract":"We consider a special variant of a pursuit-evasion game called lions and contamination. In a graph whose vertices are originally contaminated, a set of lions walk around the graph and clear the contamination from every vertex they visit. The contamination, however, simultaneously spreads to any adjacent vertex not occupied by a lion. We study the relationship between different types of clearings of graphs, such as clearings which do not allow recontamination, clearings where at most one lion moves at each time step and clearings where lions are forbidden to be stacked on the same vertex. We answer several questions raised by Adams et al. [2].","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"103 1","pages":"101961"},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73656692","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}
Thomas Bläsius, T. Friedrich, Martin S. Krejca, Louise Molitor
{"title":"The Impact of Geometry on Monochrome Regions in the Flip Schelling Process","authors":"Thomas Bläsius, T. Friedrich, Martin S. Krejca, Louise Molitor","doi":"10.4230/LIPIcs.ISAAC.2021.29","DOIUrl":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.29","url":null,"abstract":"11 Schelling’s classical segregation model gives a coherent explanation for the wide-spread phenomenon 12 of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling 13 Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the 14 predominant type in their neighborhood, decide whether to change their types; similar to a new 15 agent arriving as soon as another agent leaves the vertex. 16 We investigate the probability that an edge { u, v } is monochrome, i.e., that both vertices u and v 17 have the same type in the FSP, and we provide a general framework for analyzing the influence of 18 the underlying graph topology on residential segregation. In particular, for two adjacent vertices, 19 we show that a highly decisive common neighborhood, i.e., a common neighborhood where the 20 absolute value of the difference between the number of vertices with different types is high, supports 21 segregation and, moreover, that large common neighborhoods are more decisive. 22 As an application, we study the expected behavior of the FSP on two common random graph 23 models with and without geometry: (1) For random geometric graphs, we show that the existence of 24 an edge { u, v } makes a highly decisive common neighborhood for u and v more likely. Based on 25 this, we prove the existence of a constant c > 0 such that the expected fraction of monochrome 26 edges after the FSP is at least 1 / 2 + c . (2) For Erdős–Rényi graphs we show that large common 27 neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is 28 at most 1 / 2 + o (1). Our results indicate that the cluster structure of the underlying graph has a 29 significant impact on the obtained segregation","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"220 1","pages":"101902"},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89120243","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":"Bottleneck Matching in the Plane","authors":"M. J. Katz, M. Sharir","doi":"10.48550/arXiv.2205.05887","DOIUrl":"https://doi.org/10.48550/arXiv.2205.05887","url":null,"abstract":"We present an algorithm for computing a bottleneck matching in a set of $n=2ell$ points in the plane, which runs in $O(n^{omega/2}log n)$ deterministic time, where $omegaapprox 2.37$ is the exponent of matrix multiplication.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"87 1","pages":"101986"},"PeriodicalIF":0.0,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80134069","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}
Rachel Friederich, Matthew Graham, Anirban Ghosh, Brian Hicks, Ronald Shevchenko
{"title":"Experiments with Unit Disk Cover Algorithms for Covering Massive Pointsets","authors":"Rachel Friederich, Matthew Graham, Anirban Ghosh, Brian Hicks, Ronald Shevchenko","doi":"10.48550/arXiv.2205.01716","DOIUrl":"https://doi.org/10.48550/arXiv.2205.01716","url":null,"abstract":"Given a set of n points in the plane, the Unit Disk Cover (UDC) problem asks to compute the minimum number of unit disks required to cover the points, along with a placement of the disks. The problem is NP-hard and several approximation algorithms have been designed over the last three decades. In this paper, we have engineered and experimentally compared practical performances of some of these algorithms on massive pointsets. The goal is to investigate which algorithms run fast and give good approximation in practice. We present a simple 7-approximation algorithm for UDC that runs in O ( n ) expected time and uses O ( s ) extra space, where s denotes the size of the generated cover. In our experiments, it turned out to be the speediest of all. We also present two heuristics to reduce the sizes of covers generated by it without slowing it down by much. To our knowledge, this is the first work that experimentally compares geometric covering algorithms. Experiments with them using massive pointsets (in the order of millions) throw light on their practical uses. We share the engineered algorithms via GitHub 1 for broader uses and future research in the domain of geometric optimization.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"6 1","pages":"101925"},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85886781","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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