Euler diagrams are a tool for the graphical representation of set relations. Due to their simple way of visualizing elements in the sets by geometric containment, they are easily readable by an inexperienced reader. Euler diagrams where the sets are visualized as aligned rectangles are of special interest. In this work, we link the existence of such rectangular Euler diagrams to the order dimension of an associated order relation. For this, we consider Euler diagrams in one and two dimensions. In the one-dimensional case, this correspondence provides us with a polynomial-time algorithm to compute the Euler diagrams, while the two-dimensional case results in an exponential-time algorithm.
{"title":"Realizability of Rectangular Euler Diagrams","authors":"Dominik Dürrschnabel, Uta Priss","doi":"arxiv-2403.03801","DOIUrl":"https://doi.org/arxiv-2403.03801","url":null,"abstract":"Euler diagrams are a tool for the graphical representation of set relations.\u0000Due to their simple way of visualizing elements in the sets by geometric\u0000containment, they are easily readable by an inexperienced reader. Euler\u0000diagrams where the sets are visualized as aligned rectangles are of special\u0000interest. In this work, we link the existence of such rectangular Euler\u0000diagrams to the order dimension of an associated order relation. For this, we\u0000consider Euler diagrams in one and two dimensions. In the one-dimensional case,\u0000this correspondence provides us with a polynomial-time algorithm to compute the\u0000Euler diagrams, while the two-dimensional case results in an exponential-time\u0000algorithm.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140057233","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 give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden set number. We then give some important examples in which these quantities don't always coincide. Finally, We present some consequences of insights from Browne, Kasthurirangan, Mitchell and Polishchuk [FOCS, 2023] on other classes of simple polygons.
{"title":"An Overview of Minimum Convex Cover and Maximum Hidden Set","authors":"Reilly Browne","doi":"arxiv-2403.01354","DOIUrl":"https://doi.org/arxiv-2403.01354","url":null,"abstract":"We give a review of results on the minimum convex cover and maximum hidden\u0000set problems. In addition, we give some new results. First we show that it is\u0000NP-hard to determine whether a polygon has the same convex cover number as its\u0000hidden set number. We then give some important examples in which these\u0000quantities don't always coincide. Finally, We present some consequences of\u0000insights from Browne, Kasthurirangan, Mitchell and Polishchuk [FOCS, 2023] on\u0000other classes of simple polygons.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036024","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 study computationally-hard fundamental motion planning problems where the goal is to translate $k$ axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size, which is the number of bits needed to encode the coordinates of the rectangles' vertices. We focus on axis-aligned translations, and more generally, translations restricted to a given set of directions, and we study the two settings where the robots move in the free plane, and where they are confined to a bounding box. We obtain fixed-parameter tractable (FPT) algorithms parameterized by $k$ for all the settings under consideration. In the case where the robots move serially (i.e., one in each time step) and axis-aligned, we prove a structural result stating that every problem instance admits an optimal solution in which the moves are along a grid, whose size is a function of $k$, that can be defined based on the input instance. This structural result implies that the problem is fixed-parameter tractable parameterized by $k$. We also consider the case in which the robots move in parallel (i.e., multiple robots can move during the same time step), and which falls under the category of Coordinated Motion Planning problems. Finally, we show that, when the robots move in the free plane, the FPT results for the serial motion case carry over to the case where the translations are restricted to any given set of directions.
{"title":"On the Parameterized Complexity of Motion Planning for Rectangular Robots","authors":"Iyad Kanj, Salman Parsa","doi":"arxiv-2402.17846","DOIUrl":"https://doi.org/arxiv-2402.17846","url":null,"abstract":"We study computationally-hard fundamental motion planning problems where the\u0000goal is to translate $k$ axis-aligned rectangular robots from their initial\u0000positions to their final positions without collision, and with the minimum\u0000number of translation moves. Our aim is to understand the interplay between the\u0000number of robots and the geometric complexity of the input instance measured by\u0000the input size, which is the number of bits needed to encode the coordinates of\u0000the rectangles' vertices. We focus on axis-aligned translations, and more\u0000generally, translations restricted to a given set of directions, and we study\u0000the two settings where the robots move in the free plane, and where they are\u0000confined to a bounding box. We obtain fixed-parameter tractable (FPT)\u0000algorithms parameterized by $k$ for all the settings under consideration. In\u0000the case where the robots move serially (i.e., one in each time step) and\u0000axis-aligned, we prove a structural result stating that every problem instance\u0000admits an optimal solution in which the moves are along a grid, whose size is a\u0000function of $k$, that can be defined based on the input instance. This\u0000structural result implies that the problem is fixed-parameter tractable\u0000parameterized by $k$. We also consider the case in which the robots move in\u0000parallel (i.e., multiple robots can move during the same time step), and which\u0000falls under the category of Coordinated Motion Planning problems. Finally, we\u0000show that, when the robots move in the free plane, the FPT results for the\u0000serial motion case carry over to the case where the translations are restricted\u0000to any given set of directions.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140004449","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}
Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set $S$ of $n$ colored points in $mathbb{R}^d$, which implicitly defines a graph $G = (S,E(S))$ where $E(S) = {(p,q): p,q in S text{ have different colors}}$, and the goal is to compute a minimum-cost subset $E^* subseteq E(S)$ of edges that cover all points in $S$. Here the cost of $E^*$ is the sum of the costs of all edges in $E^*$, where the cost of a single edge $e$ is the Euclidean distance (or more generally, the $L_p$-distance) between the two endpoints of $e$. Our main result is a $(1+varepsilon)$-approximation algorithm with an optimal running time $O_varepsilon(n log n)$ for geometric many-to-many matching in any fixed dimension, which works under any $L_p$-norm. This is the first near-linear approximation scheme for the problem in any $d geq 2$. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in $mathbb{R}^1$ and $mathbb{R}^2$, and the best known approximation scheme in $mathbb{R}^2$ takes $O_varepsilon(n^{1.5} cdot mathsf{poly}(log n))$ time.
{"title":"An $O(n log n)$-Time Approximation Scheme for Geometric Many-to-Many Matching","authors":"Sayan Bandyapadhyay, Jie Xue","doi":"arxiv-2402.15837","DOIUrl":"https://doi.org/arxiv-2402.15837","url":null,"abstract":"Geometric matching is an important topic in computational geometry and has\u0000been extensively studied over decades. In this paper, we study a\u0000geometric-matching problem, known as geometric many-to-many matching. In this\u0000problem, the input is a set $S$ of $n$ colored points in $mathbb{R}^d$, which\u0000implicitly defines a graph $G = (S,E(S))$ where $E(S) = {(p,q): p,q in S\u0000text{ have different colors}}$, and the goal is to compute a minimum-cost\u0000subset $E^* subseteq E(S)$ of edges that cover all points in $S$. Here the\u0000cost of $E^*$ is the sum of the costs of all edges in $E^*$, where the cost of\u0000a single edge $e$ is the Euclidean distance (or more generally, the\u0000$L_p$-distance) between the two endpoints of $e$. Our main result is a\u0000$(1+varepsilon)$-approximation algorithm with an optimal running time\u0000$O_varepsilon(n log n)$ for geometric many-to-many matching in any fixed\u0000dimension, which works under any $L_p$-norm. This is the first near-linear\u0000approximation scheme for the problem in any $d geq 2$. Prior to this work,\u0000only the bipartite case of geometric many-to-many matching was considered in\u0000$mathbb{R}^1$ and $mathbb{R}^2$, and the best known approximation scheme in\u0000$mathbb{R}^2$ takes $O_varepsilon(n^{1.5} cdot mathsf{poly}(log n))$ time.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978864","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 give an embedding of the Poincar'e halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(log D)$. We also show how to construct, for any $k$, an $O(klog D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n cdot lambda_k(n)$ edges, where $lambda_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we present a data structure for approximate near-neighbor searching in $H^D$, with construction time $2^{O(D)}nlog n$, query time $2^{O(D)}log n$ and additive error $O(log D)$. These constructions can be done in $2^{O(D)}n log n$ time.
{"title":"Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces","authors":"Eunku Park, Antoine Vigneron","doi":"arxiv-2402.14604","DOIUrl":"https://doi.org/arxiv-2402.14604","url":null,"abstract":"We give an embedding of the Poincar'e halfspace $H^D$ into a discrete metric\u0000space based on a binary tiling of $H^D$, with additive distortion $O(log D)$.\u0000It yields the following results. We show that any subset $P$ of $n$ points in\u0000$H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges,\u0000and with additive distortion $O(log D)$. We also show how to construct, for\u0000any $k$, an $O(klog D)$-purely additive spanner of $P$ with $2^{O(D)}n$\u0000Steiner vertices and $2^{O(D)}n cdot lambda_k(n)$ edges, where $lambda_k(n)$\u0000is the $k$th-row inverse Ackermann function. Finally, we present a data\u0000structure for approximate near-neighbor searching in $H^D$, with construction\u0000time $2^{O(D)}nlog n$, query time $2^{O(D)}log n$ and additive error $O(log\u0000D)$. These constructions can be done in $2^{O(D)}n log n$ time.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948833","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}
Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, Ignaz Rutter
We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem: given a plane drawing of a planar graph $G$ and a set of edges $F$, insert the edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In this paper, we focus on the $1$-PIP scenario. We present a linear-time algorithm for the case that $G$ is a triangulation, while proving NP-completeness for the case that $G$ is biconnected and $F$ forms a path or a matching.
{"title":"On $k$-Plane Insertion into Plane Drawings","authors":"Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, Ignaz Rutter","doi":"arxiv-2402.14552","DOIUrl":"https://doi.org/arxiv-2402.14552","url":null,"abstract":"We introduce the $k$-Plane Insertion into Plane drawing ($k$-PIP) problem:\u0000given a plane drawing of a planar graph $G$ and a set of edges $F$, insert the\u0000edges in $F$ into the drawing such that the resulting drawing is $k$-plane. In\u0000this paper, we focus on the $1$-PIP scenario. We present a linear-time\u0000algorithm for the case that $G$ is a triangulation, while proving\u0000NP-completeness for the case that $G$ is biconnected and $F$ forms a path or a\u0000matching.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948826","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 study the Generalized Red-Blue Annulus Cover problem for two sets of points, red ($R$) and blue ($B$), where each point $p in Rcup B$ is associated with a positive penalty ${cal P}(p)$. The red points have non-covering penalties, and the blue points have covering penalties. The objective is to compute a circular annulus ${cal A}$ such that the value of the function ${cal P}({R}^{out})$ + ${cal P}({ B}^{in})$ is minimum, where ${R}^{out} subseteq {R}$ is the set of red points not covered by ${cal A}$ and ${B}^{in} subseteq {B}$ is the set of blue points covered by $cal A$. We also study another version of this problem, where all the red points in $R$ and the minimum number of points in $B$ are covered by the circular annulus in two dimensions. We design polynomial-time algorithms for all such circular annulus problems.
{"title":"Generalized Red-Blue Circular Annulus Cover Problem","authors":"Sukanya Maji, Supantha Pandit, Sanjib Sadhu","doi":"arxiv-2402.13767","DOIUrl":"https://doi.org/arxiv-2402.13767","url":null,"abstract":"We study the Generalized Red-Blue Annulus Cover problem for two sets of\u0000points, red ($R$) and blue ($B$), where each point $p in Rcup B$ is\u0000associated with a positive penalty ${cal P}(p)$. The red points have\u0000non-covering penalties, and the blue points have covering penalties. The\u0000objective is to compute a circular annulus ${cal A}$ such that the value of\u0000the function ${cal P}({R}^{out})$ + ${cal P}({ B}^{in})$ is minimum, where\u0000${R}^{out} subseteq {R}$ is the set of red points not covered by ${cal A}$\u0000and ${B}^{in} subseteq {B}$ is the set of blue points covered by $cal A$. We\u0000also study another version of this problem, where all the red points in $R$ and\u0000the minimum number of points in $B$ are covered by the circular annulus in two\u0000dimensions. We design polynomial-time algorithms for all such circular annulus\u0000problems.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919845","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}
Brittany Terese Fasy, David L. Millman, Anna Schenfisch
Recent developments in shape reconstruction and comparison call for the use of many different types of topological descriptors (persistence diagrams, Euler characteristic functions, etc.). We establish a framework that allows for quantitative comparisons of topological descriptor types and therefore may be used as a tool in more rigorously justifying choices made in applications. We then use this framework to partially order a set of six common topological descriptor types. In particular, the resulting poset gives insight into the advantages of using verbose rather than concise topological descriptors. We then provide lower bounds on the size of sets of descriptors that are complete discrete invariants of simplicial complexes, both tight and worst case. This work sets up a rigorous theory that allows for future comparisons and analysis of topological descriptor types.
{"title":"Ordering Topological Descriptors","authors":"Brittany Terese Fasy, David L. Millman, Anna Schenfisch","doi":"arxiv-2402.13632","DOIUrl":"https://doi.org/arxiv-2402.13632","url":null,"abstract":"Recent developments in shape reconstruction and comparison call for the use\u0000of many different types of topological descriptors (persistence diagrams, Euler\u0000characteristic functions, etc.). We establish a framework that allows for\u0000quantitative comparisons of topological descriptor types and therefore may be\u0000used as a tool in more rigorously justifying choices made in applications. We\u0000then use this framework to partially order a set of six common topological\u0000descriptor types. In particular, the resulting poset gives insight into the\u0000advantages of using verbose rather than concise topological descriptors. We\u0000then provide lower bounds on the size of sets of descriptors that are complete\u0000discrete invariants of simplicial complexes, both tight and worst case. This\u0000work sets up a rigorous theory that allows for future comparisons and analysis\u0000of topological descriptor types.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919764","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}
Ivor van der Hoog, Thijs van der Horst, Tim Ophelders
Given a trajectory $T$ and a distance $Delta$, we wish to find a set $C$ of curves of complexity at most $ell$, such that we can cover $T$ with subcurves that each are within Fr'echet distance $Delta$ to at least one curve in $C$. We call $C$ an $(ell,Delta)$-clustering and aim to find an $(ell,Delta)$-clustering of minimum cardinality. This problem was introduced by Akitaya $et$ $al.$ (2021) and shown to be NP-complete. The main focus has therefore been on bicriterial approximation algorithms, allowing for the clustering to be an $(ell, Theta(Delta))$-clustering of roughly optimal size. We present algorithms that construct $(ell,4Delta)$-clusterings of $mathcal{O}(k log n)$ size, where $k$ is the size of the optimal $(ell, Delta)$-clustering. For the discrete Fr'echet distance, we use $mathcal{O}(n ell log n)$ space and $mathcal{O}(k n^2 log^3 n)$ deterministic worst case time. For the continuous Fr'echet distance, we use $mathcal{O}(n^2 log n)$ space and $mathcal{O}(k n^3 log^3 n)$ time. Our algorithms significantly improve upon the clustering quality (improving the approximation factor in $Delta$) and size (whenever $ell in Omega(log n)$). We offer deterministic running times comparable to known expected bounds. Additionally, in the continuous setting, we give a near-linear improvement upon the space usage. When compared only to deterministic results, we offer a near-linear speedup and a near-quadratic improvement in the space usage. When we may restrict ourselves to only considering clusters where all subtrajectories are vertex-to-vertex subcurves, we obtain even better results under the continuous Fr'echet distance. Our algorithm becomes near quadratic and uses space that is near linear in $n ell$.
{"title":"Faster and Deterministic Subtrajectory Clustering","authors":"Ivor van der Hoog, Thijs van der Horst, Tim Ophelders","doi":"arxiv-2402.13117","DOIUrl":"https://doi.org/arxiv-2402.13117","url":null,"abstract":"Given a trajectory $T$ and a distance $Delta$, we wish to find a set $C$ of\u0000curves of complexity at most $ell$, such that we can cover $T$ with subcurves\u0000that each are within Fr'echet distance $Delta$ to at least one curve in $C$.\u0000We call $C$ an $(ell,Delta)$-clustering and aim to find an\u0000$(ell,Delta)$-clustering of minimum cardinality. This problem was introduced\u0000by Akitaya $et$ $al.$ (2021) and shown to be NP-complete. The main focus has\u0000therefore been on bicriterial approximation algorithms, allowing for the\u0000clustering to be an $(ell, Theta(Delta))$-clustering of roughly optimal\u0000size. We present algorithms that construct $(ell,4Delta)$-clusterings of\u0000$mathcal{O}(k log n)$ size, where $k$ is the size of the optimal $(ell,\u0000Delta)$-clustering. For the discrete Fr'echet distance, we use $mathcal{O}(n\u0000ell log n)$ space and $mathcal{O}(k n^2 log^3 n)$ deterministic worst case\u0000time. For the continuous Fr'echet distance, we use $mathcal{O}(n^2 log n)$\u0000space and $mathcal{O}(k n^3 log^3 n)$ time. Our algorithms significantly\u0000improve upon the clustering quality (improving the approximation factor in\u0000$Delta$) and size (whenever $ell in Omega(log n)$). We offer deterministic\u0000running times comparable to known expected bounds. Additionally, in the\u0000continuous setting, we give a near-linear improvement upon the space usage.\u0000When compared only to deterministic results, we offer a near-linear speedup and\u0000a near-quadratic improvement in the space usage. When we may restrict ourselves\u0000to only considering clusters where all subtrajectories are vertex-to-vertex\u0000subcurves, we obtain even better results under the continuous Fr'echet\u0000distance. Our algorithm becomes near quadratic and uses space that is near\u0000linear in $n ell$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920033","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}
Steven van den Broek, Wouter Meulemans, Bettina Speckmann
Constructing partitions of colored points is a well-studied problem in discrete and computational geometry. We study the problem of creating a minimum-cardinality partition into monochromatic islands. Our input is a set $S$ of $n$ points in the plane where each point has one of $k geq 2$ colors. A set of points is monochromatic if it contains points of only one color. An island $I$ is a subset of $S$ such that $mathcal{CH}(I) cap S = I$, where $mathcal{CH}(I)$ denotes the convex hull of $I$. We identify an island with its convex hull; therefore, a partition into islands has the additional requirement that the convex hulls of the islands are pairwise-disjoint. We present three greedy algorithms for constructing island partitions and analyze their approximation ratios.
构造彩色点的分区是离散几何和计算几何中一个研究得很透彻的问题。我们研究的问题是将最小心率分割为单色岛。我们的输入是由平面上 $n$ 点组成的集合$S$，其中每个点都有 $k geq 2$ 种颜色。如果点集合只包含一种颜色的点，那么它就是单色的。岛屿 $I$ 是 $S$ 的一个子集，使得 $mathcal{CH}(I) cap S = I$，其中$mathcal{CH}(I)$ 表示 $I$ 的凸壳。我们将一个岛与它的凸壳进行标识；因此，将一个岛分割成多个岛还有一个额外的要求，即岛的凸壳必须是成对相交的。我们提出了三种构建岛屿分割的贪婪算法，并分析了它们的近似率。
{"title":"Greedy Monochromatic Island Partitions","authors":"Steven van den Broek, Wouter Meulemans, Bettina Speckmann","doi":"arxiv-2402.13340","DOIUrl":"https://doi.org/arxiv-2402.13340","url":null,"abstract":"Constructing partitions of colored points is a well-studied problem in\u0000discrete and computational geometry. We study the problem of creating a\u0000minimum-cardinality partition into monochromatic islands. Our input is a set\u0000$S$ of $n$ points in the plane where each point has one of $k geq 2$ colors. A\u0000set of points is monochromatic if it contains points of only one color. An\u0000island $I$ is a subset of $S$ such that $mathcal{CH}(I) cap S = I$, where\u0000$mathcal{CH}(I)$ denotes the convex hull of $I$. We identify an island with\u0000its convex hull; therefore, a partition into islands has the additional\u0000requirement that the convex hulls of the islands are pairwise-disjoint. We\u0000present three greedy algorithms for constructing island partitions and analyze\u0000their approximation ratios.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139919780","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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