Pub Date : 2025-04-15DOI: 10.1016/j.comgeo.2025.102195
Michael A. Bekos, Charis Papadopoulos
{"title":"CGTA","authors":"Michael A. Bekos, Charis Papadopoulos","doi":"10.1016/j.comgeo.2025.102195","DOIUrl":"10.1016/j.comgeo.2025.102195","url":null,"abstract":"","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102195"},"PeriodicalIF":0.4,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143842584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-09DOI: 10.1016/j.comgeo.2025.102194
Jacobus Conradi , Anne Driemel , Benedikt Kolbe
Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in .
We assume that the algorithm is given two continuous curves, each consisting of a sequence of m, resp. n, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in time, leading to a computation of the Fréchet distance that runs in time.
Furthermore, we study approximation algorithms for piecewise smooth curves that are also c-packed for some fixed value c. We adapt the existing framework for polygonal curves that leads to a near-linear -approximation to the Fréchet distance to the setting of piecewise smooth curves.
{"title":"Revisiting the Fréchet distance between piecewise smooth curves","authors":"Jacobus Conradi , Anne Driemel , Benedikt Kolbe","doi":"10.1016/j.comgeo.2025.102194","DOIUrl":"10.1016/j.comgeo.2025.102194","url":null,"abstract":"<div><div>Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div><div>We assume that the algorithm is given two continuous curves, each consisting of a sequence of <em>m</em>, resp. <em>n</em>, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo></math></span> time, leading to a computation of the Fréchet distance that runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mi>log</mi><mo></mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time.</div><div>Furthermore, we study approximation algorithms for piecewise smooth curves that are also <em>c</em>-packed for some fixed value <em>c</em>. We adapt the existing framework for polygonal curves that leads to a near-linear <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation to the Fréchet distance to the setting of piecewise smooth curves.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102194"},"PeriodicalIF":0.4,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-27DOI: 10.1016/j.comgeo.2025.102192
Emilio Di Giacomo , Walter Didimo , Giuseppe Liotta , Henk Meijer , Fabrizio Montecchiani , Stephen Wismath
The edge-length ratio of a planar straight-line drawing Γ of a graph G is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the n-vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.
{"title":"Bounds on the edge-length ratio of 2-outerplanar graphs","authors":"Emilio Di Giacomo , Walter Didimo , Giuseppe Liotta , Henk Meijer , Fabrizio Montecchiani , Stephen Wismath","doi":"10.1016/j.comgeo.2025.102192","DOIUrl":"10.1016/j.comgeo.2025.102192","url":null,"abstract":"<div><div>The edge-length ratio of a planar straight-line drawing Γ of a graph <em>G</em> is the largest ratio between the lengths of every pair of edges of Γ. If the ratio is measured by considering only pairs of edges that are incident to a common vertex, we talk about local edge-length ratio. The (local) edge-length ratio of a planar graph is the infimum over all (local) edge-length ratios of its planar straight-line drawings. It is known that the edge-length ratio of outerplanar graphs is upper bounded by a constant, while there exist graph families with non-constant outerplanarity that have non-constant lower bounds on their edge-length ratios. In this paper we prove an <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> lower bound on the local edge-length ratio (and hence on the edge-length ratio) of the <em>n</em>-vertex 2-outerplanar graphs. We also prove a constant upper bound on the edge-length ratio of Halin graphs, pseudo-Halin graphs, and their generalizations.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102192"},"PeriodicalIF":0.4,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-26DOI: 10.1016/j.comgeo.2025.102191
Thomas Depian , Martin Nöllenburg , Soeren Terziadis , Markus Wallinger
Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce grouping and ordering constraints in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is NP-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is NP-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.
{"title":"Constrained boundary labeling","authors":"Thomas Depian , Martin Nöllenburg , Soeren Terziadis , Markus Wallinger","doi":"10.1016/j.comgeo.2025.102191","DOIUrl":"10.1016/j.comgeo.2025.102191","url":null,"abstract":"<div><div>Boundary labeling is a technique in computational geometry used to label sets of features in an illustration. It involves placing labels along an axis-parallel bounding box and connecting each label with its corresponding feature using non-crossing leader lines. Although boundary labeling is well-studied, semantic constraints on the labels have not been investigated thoroughly. In this paper, we introduce <em>grouping</em> and <em>ordering constraints</em> in boundary labeling: Grouping constraints enforce that all labels in a group are placed consecutively on the boundary, and ordering constraints enforce a partial order over the labels. We show that it is <span>NP</span>-hard to find a labeling for arbitrarily sized labels with unrestricted positions along one side of the boundary. However, we obtain polynomial-time algorithms if we restrict this problem either to uniform-height labels or to a finite set of candidate positions. Furthermore, we show that finding a labeling on two opposite sides of the boundary is <span>NP</span>-complete, even for uniform-height labels and finite label positions. Finally, we experimentally confirm that our approach has also practical relevance.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102191"},"PeriodicalIF":0.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-26DOI: 10.1016/j.comgeo.2025.102193
Ji Hoon Chun, Christian Kipp, Sandro Roch
We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an exact cover is not possible.
{"title":"On exact covering with unit disks","authors":"Ji Hoon Chun, Christian Kipp, Sandro Roch","doi":"10.1016/j.comgeo.2025.102193","DOIUrl":"10.1016/j.comgeo.2025.102193","url":null,"abstract":"<div><div>We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an exact cover is not possible.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102193"},"PeriodicalIF":0.4,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-25DOI: 10.1016/j.comgeo.2025.102190
Samantha Chen, Yusu Wang
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.
{"title":"Approximation algorithms for 1-Wasserstein distance between persistence diagrams","authors":"Samantha Chen, Yusu Wang","doi":"10.1016/j.comgeo.2025.102190","DOIUrl":"10.1016/j.comgeo.2025.102190","url":null,"abstract":"<div><div>Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102190"},"PeriodicalIF":0.4,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143738783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-19DOI: 10.1016/j.comgeo.2025.102189
Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen
Given a set of autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small -sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in rounds with probability at least . The algorithms are collision-free and do not require the knowledge of the number of robots.
{"title":"Pattern formation for fat robots with memory","authors":"Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen","doi":"10.1016/j.comgeo.2025.102189","DOIUrl":"10.1016/j.comgeo.2025.102189","url":null,"abstract":"<div><div>Given a set of <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span> autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> rounds with probability at least <span><math><mn>1</mn><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>q</mi></mrow></msup></math></span>. The algorithms are collision-free and do not require the knowledge of the number of robots.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102189"},"PeriodicalIF":0.4,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-18DOI: 10.1016/j.comgeo.2025.102185
Chaeyoon Chung , Taehoon Ahn , Sang Won Bae , Hee-Kap Ahn
Given a set P of n points in the plane and an integer , the k-line-center problem asks k slabs whose union covers P that minimizes the maximum width of the k slabs. In this paper, we introduce a new variant of the k-line-center problem for , in which the resulting k lines are parallel and a prescribed separation between two line centers is guaranteed. More precisely, we define a measure of separation, namely the gap-ratio of k parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the k slabs. We present efficient algorithms for the following problems: (1) Given a real , compute k parallel slabs of minimum width that cover P with gap-ratio at least ρ. (2) Compute k parallel slabs that cover P with maximum possible gap-ratio. Our algorithms run in and time, respectively, using space, where denotes the maximum possible gap-ratio of any k parallel slabs that cover P. Using linear space, the running times only slightly increase to and .
{"title":"Parallel line centers with guaranteed separation","authors":"Chaeyoon Chung , Taehoon Ahn , Sang Won Bae , Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102185","DOIUrl":"10.1016/j.comgeo.2025.102185","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points in the plane and an integer <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, the <em>k</em>-line-center problem asks <em>k</em> slabs whose union covers <em>P</em> that minimizes the maximum width of the <em>k</em> slabs. In this paper, we introduce a new variant of the <em>k</em>-line-center problem for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, in which the resulting <em>k</em> lines are parallel and a prescribed separation between two line centers is guaranteed. More precisely, we define a measure of separation, namely the gap-ratio of <em>k</em> parallel slabs, to be the minimum distance between any two slabs, divided by the width of the smallest slab enclosing the <em>k</em> slabs. We present efficient algorithms for the following problems: (1) Given a real <span><math><mn>0</mn><mo><</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></math></span>, compute <em>k</em> parallel slabs of minimum width that cover <em>P</em> with gap-ratio at least <em>ρ</em>. (2) Compute <em>k</em> parallel slabs that cover <em>P</em> with maximum possible gap-ratio. Our algorithms run in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>+</mo><mi>k</mi><mi>n</mi><mo>)</mo><mo>)</mo></math></span> time, respectively, using <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo></mo><mi>k</mi><mo>)</mo></math></span> space, where <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span> denotes the maximum possible gap-ratio of any <em>k</em> parallel slabs that cover <em>P</em>. Using linear space, the running times only slightly increase to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msubsup><mo>⋅</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102185"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-18DOI: 10.1016/j.comgeo.2025.102188
Gang Liu, Haitao Wang
Given a set P of n points and a set S of n weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line ℓ. We present an time algorithm for the problem. This improves the previously best work of time. Our result leads to an algorithm of time for the halfplane coverage problem (i.e., using n weighted halfplanes to cover n points), an improvement over the previous time solution. If all halfplanes are lower ones, our algorithm runs in time, while the previous best algorithm takes time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.
{"title":"On line-separable weighted unit-disk coverage and related problems","authors":"Gang Liu, Haitao Wang","doi":"10.1016/j.comgeo.2025.102188","DOIUrl":"10.1016/j.comgeo.2025.102188","url":null,"abstract":"<div><div>Given a set <em>P</em> of <em>n</em> points and a set <em>S</em> of <em>n</em> weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of <em>P</em>. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of <em>P</em> by a line <em>ℓ</em>. We present an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></math></span> time algorithm for the problem. This improves the previously best work of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time. Our result leads to an algorithm of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></math></span> time for the halfplane coverage problem (i.e., using <em>n</em> weighted halfplanes to cover <em>n</em> points), an improvement over the previous <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time solution. If all halfplanes are lower ones, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></math></span> time, while the previous best algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102188"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-18DOI: 10.1016/j.comgeo.2025.102186
János Pach , Morteza Saghafian , Patrick Schnider
We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.
{"title":"Decomposition of geometric graphs into star-forests","authors":"János Pach , Morteza Saghafian , Patrick Schnider","doi":"10.1016/j.comgeo.2025.102186","DOIUrl":"10.1016/j.comgeo.2025.102186","url":null,"abstract":"<div><div>We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on <em>n</em> vertices cannot be decomposed into fewer than <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102186"},"PeriodicalIF":0.4,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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