Pub Date : 2023-09-01DOI: 10.4230/LIPIcs.ISAAC.2022.42
Pankaj Agarwal, M. J. Katz, M. Sharir
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
设S为r2中n个等复杂度几何对象(点、线段、圆盘、椭圆)的集合，设ϱ: S × S→R≥0为S上的距离函数。当参数r≥0时，定义邻近图G (r) = (S, E)，其中E = {(E 1, E 2)∈S × S | E 1 ε = e2， ϱ (E 1, e2)≤r}。给定S, S, t∈S，且整数k≥1，逆最短路径(RSP)问题要求计算r∗≥0的最小值，使得G (r∗)包含从S到t的最长长度为k的路径。在本文中，我们提出了一种通用的随机化技术，可以有效地解决大量几何对象和距离函数的RSP问题。使用标准的，有时更复杂的，半代数范围搜索技术，我们首先给出了决策问题的一个有效算法，即给定值r≥0，确定G (r)是否包含从s到t的最长长度为k的路径。接下来，我们调整我们的决策算法，并将其与随机抽样方法相结合，通过在包含r∗的O (n 2)个候选值的隐式集合上有效地执行二分搜索来计算r∗。我们通过将一般技术应用于各种几何接近图来说明它的多功能性。例如，我们得到(i)一个O∗(n 4 / 3)期望时间随机化算法(其中O∗(·)隐藏了多对数(n)个因子)，其中S是r2中一对不相交的线段集合，并且ϱ (e 1, e 2) = min x∈e 1,y∈e 2∥x−y∥(其中∥·∥是欧几里得距离)，并且(ii)
{"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}
Pub Date : 2023-05-01DOI: 10.1007/978-3-030-83508-8_33
Joachim Gudmundsson, Y. Sha
{"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}
Pub Date : 2023-03-01DOI: 10.4230/LIPIcs.ISAAC.2021.45
Joachim Gudmundsson, Y. Sha, Fan Yao
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.
{"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}
Pub Date : 2023-01-01DOI: 10.4230/LIPIcs.ISAAC.2021.46
K. Cho, Eunjin Oh
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 .
{"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}
Pub Date : 2022-11-01DOI: 10.1007/978-3-030-79987-8_28
Byeonguk Kang, J. Choi, Hee-Kap Ahn
{"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}
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.
{"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}
Pub Date : 2022-10-01DOI: 10.4230/LIPIcs.SWAT.2022.17
Daniel Bertschinger, Meghana M. Reddy, Enrico Mann
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].
{"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}
Pub Date : 2022-06-01DOI: 10.4230/LIPIcs.ISAAC.2021.29
Thomas Bläsius, T. Friedrich, Martin S. Krejca, Louise Molitor
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
{"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}
Pub Date : 2022-05-12DOI: 10.48550/arXiv.2205.05887
M. J. Katz, M. Sharir
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.
{"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}
Pub Date : 2022-05-03DOI: 10.48550/arXiv.2205.01716
Rachel Friederich, Matthew Graham, Anirban Ghosh, Brian Hicks, Ronald Shevchenko
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.
给定平面上的n个点，单位磁盘覆盖(Unit Disk Cover, UDC)问题要求计算覆盖这些点所需的最小单位磁盘数量，以及磁盘的位置。这个问题是np困难的，在过去的三十年里已经设计了几种近似算法。在本文中，我们设计并实验比较了其中一些算法在大量点集上的实际性能。目的是研究哪些算法在实践中运行速度快并给出良好的近似。我们为UDC提供了一个简单的7近似算法，该算法在O (n)个预期时间内运行，并使用O (s)个额外空间，其中s表示生成的覆盖的大小。在我们的实验中，它被证明是最快的。我们还提出了两种启发式方法来减少由它生成的覆盖的大小，而不会减慢它的速度。据我们所知，这是第一个实验比较几何覆盖算法的工作。使用大量的点集(数以百万计)对它们进行的实验揭示了它们的实际用途。我们通过GitHub 1分享工程算法，以便在几何优化领域进行更广泛的应用和未来的研究。
{"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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