Computational Geometry-Theory and Applications最新文献

Approximation algorithms for 1-Wasserstein distance between persistence diagrams

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-25 DOI: 10.1016/j.comgeo.2025.102190

Samantha Chen, Yusu Wang

{"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}

引用次数: 0

Pattern formation for fat robots with memory

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-19 DOI: 10.1016/j.comgeo.2025.102189

Rusul J. Alsaedi, Joachim Gudmundsson, André van Renssen

{"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 <math><mi>n</mi><mo>≥</mo><mn>1</mn></math> 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 <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>-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 <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> rounds with probability at least <math><mn>1</mn><mo>−</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>q</mi></mrow></msup></math>. 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}

引用次数: 0

Parallel line centers with guaranteed separation

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-18 DOI: 10.1016/j.comgeo.2025.102185

Chaeyoon Chung , Taehoon Ahn , Sang Won Bae , Hee-Kap Ahn

{"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 P of n points in the plane and an integer <math><mi>k</mi><mo>≥</mo><mn>1</mn></math>, 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 <math><mi>k</mi><mo>≥</mo><mn>2</mn></math>, 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 <math><mn>0</mn><mo><</mo><mi>ρ</mi><mo>≤</mo><mn>1</mn></math>, 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 <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> and <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> time, respectively, using <math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></math> space, where <math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>max</mi></mrow></msub></math> denotes the maximum possible gap-ratio of any k parallel slabs that cover P. Using linear space, the running times only slightly increase to <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> and <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>.</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}

引用次数: 0

On line-separable weighted unit-disk coverage and related problems

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-18 DOI: 10.1016/j.comgeo.2025.102188

Gang Liu, Haitao Wang

{"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 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 <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> time algorithm for the problem. This improves the previously best work of <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> time. Our result leads to an algorithm of <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> time for the halfplane coverage problem (i.e., using n weighted halfplanes to cover n points), an improvement over the previous <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> time solution. If all halfplanes are lower ones, our algorithm runs in <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> time, while the previous best algorithm takes <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> 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}

引用次数: 0

Geometric TSP on sets

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-18 DOI: 10.1016/j.comgeo.2025.102187

Henk Alkema, Mark de Berg

引用次数: 0

Flips in odd matchings

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-03-13 DOI: 10.1016/j.comgeo.2025.102184

Oswin Aichholzer , Anna Brötzner , Daniel Perz , Patrick Schnider

引用次数: 0

Connected matchings

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-02-26 DOI: 10.1016/j.comgeo.2025.102174

Oswin Aichholzer , Sergio Cabello , Viola Mészáros , Patrick Schnider , Jan Soukup

引用次数: 0

Minimum-width double-slabs and widest empty slabs in high dimensions

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-02-25 DOI: 10.1016/j.comgeo.2025.102173

Taehoon Ahn , Chaeyoon Chung , Hee-Kap Ahn , Sang Won Bae , Otfried Cheong , Sang Duk Yoon

{"title":"Minimum-width double-slabs and widest empty slabs in high dimensions","authors":"Taehoon Ahn , Chaeyoon Chung , Hee-Kap Ahn , Sang Won Bae , Otfried Cheong , Sang Duk Yoon","doi":"10.1016/j.comgeo.2025.102173","DOIUrl":"10.1016/j.comgeo.2025.102173","url":null,"abstract":"<div><div>A slab in d-dimensional space <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> is the set of points enclosed by two parallel hyperplanes. We consider the problem of finding an optimal pair of parallel slabs, called a double-slab, that covers a given set P of n points in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>. We address two optimization problems in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> for any fixed dimension <math><mi>d</mi><mo>⩾</mo><mn>3</mn></math>: the minimum-width double-slab problem, in which one wants to minimize the maximum width of the two slabs of the resulting double-slab, and the widest empty slab problem, in which one wants to maximize the gap between the two slabs. Our results include the first nontrivial exact algorithms that solve the former problem for <math><mi>d</mi><mo>⩾</mo><mn>3</mn></math> and the latter problem for <math><mi>d</mi><mo>⩾</mo><mn>4</mn></math>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"129 ","pages":"Article 102173"},"PeriodicalIF":0.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550936","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}

引用次数: 0

Approximating average bounded-angle minimum spanning trees

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-02-18 DOI: 10.1016/j.comgeo.2025.102172

Ahmad Biniaz , Prosenjit Bose , Patrick Devaney

{"title":"Approximating average bounded-angle minimum spanning trees","authors":"Ahmad Biniaz , Prosenjit Bose , Patrick Devaney","doi":"10.1016/j.comgeo.2025.102172","DOIUrl":"10.1016/j.comgeo.2025.102172","url":null,"abstract":"<div><div>Motivated by the problem of orienting directional antennas in wireless communication networks, we study average bounded-angle minimum spanning trees. Let P be a set of points in the plane and let α be an angle. An α-spanning tree (α-ST) of P is a spanning tree of the complete Euclidean graph induced by P such that all edges incident to each point <math><mi>p</mi><mo>∈</mo><mi>P</mi></math> lie in a fixed wedge of angle α with apex p. An α-minimum spanning tree (α-MST) of P is an α-ST with minimum total edge length.</div><div>An average-α-spanning tree (denoted by <math><mover><mrow><mi>α</mi></mrow><mo>‾</mo></mover></math>-ST) is a spanning tree with the relaxed condition that incident edges to all points lie in wedges with average angle α. An average-α-minimum spanning tree (<math><mover><mrow><mi>α</mi></mrow><mo>‾</mo></mover></math>-MST) is an <math><mover><mrow><mi>α</mi></mrow><mo>‾</mo></mover></math>-ST with minimum total edge length.</div><div>Let <math><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></math> be the smallest ratio of the length of the <math><mover><mrow><mi>α</mi></mrow><mo>‾</mo></mover></math>-MST to the length of the standard MST, over all sets of points in the plane. We investigate bounds for <math><mi>A</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></math>. For <math><mi>α</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math>, Biniaz, Bose, Lubiw, and Maheshwari (Algorithmica 2022) showed that <math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>≤</mo><mi>A</mi><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>. We improve the upper bound and show that <math><mi>A</mi><mrow><mo>(</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>13</mn></mrow><mrow><mn>9</mn></mrow></mfrac></math>. We also study this for <math><mi>α</mi><mo>=</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math> and prove that <math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>≤</mo><mi>A</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>≤</mo><mn>4</mn></math>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"128 ","pages":"Article 102172"},"PeriodicalIF":0.4,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455144","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}

引用次数: 0

Finding a largest-area triangle in a terrain in near-linear time

IF 0.4 4区计算机科学

Computational Geometry-Theory and Applications Pub Date : 2025-02-17 DOI: 10.1016/j.comgeo.2025.102171

Sergio Cabello , Arun Kumar Das , Sandip Das , Joydeep Mukherjee

引用次数: 0