arXiv - CS - Computational Geometry最新文献_第9页

Constrained and Ordered Level Planarity Parameterized by the Number of Levels 以层级数为参数的受限和有序层级平面性

arXiv - CS - Computational Geometry Pub Date : 2024-03-20 DOI: arxiv-2403.13702

Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, Johannes Zink

{"title":"Constrained and Ordered Level Planarity Parameterized by the Number of Levels","authors":"Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, Johannes Zink","doi":"arxiv-2403.13702","DOIUrl":"https://doi.org/arxiv-2403.13702","url":null,"abstract":"The problem Level Planarity asks for a crossing-free drawing of a graph in\u0000the plane such that vertices are placed at prescribed y-coordinates (called\u0000levels) and such that every edge is realized as a y-monotone curve. In the\u0000variant Constrained Level Planarity (CLP), each level $y$ is equipped with a\u0000partial order $prec_y$ on its vertices and in the desired drawing the\u0000left-to-right order of vertices on level $y$ has to be a linear extension of\u0000$prec_y$. Ordered Level Planarity (OLP) corresponds to the special case of CLP\u0000where the given partial orders $prec_y$ are total orders. Previous results by\u0000Br\"uckner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019]\u0000state that both CLP and OLP are NP-hard even in severely restricted cases. In\u0000particular, they remain NP-hard even when restricted to instances whose width\u0000(the maximum number of vertices that may share a common level) is at most two.\u0000In this paper, we focus on the other dimension: we study the parameterized\u0000complexity of CLP and OLP with respect to the height (the number of levels). We show that OLP parameterized by the height is complete with respect to the\u0000complexity class XNLP, which was first studied by Elberfeld et al.\u0000[Algorithmica 2015] (under a different name) and recently made more prominent\u0000by Bodlaender et al. [FOCS 2021]. It contains all parameterized problems that\u0000can be solved nondeterministically in time $f(k) n^{O(1)}$ and space $f(k) log\u0000n$ (where $f$ is a computable function, $n$ is the input size, and $k$ is the\u0000parameter). If a problem is XNLP-complete, it lies in XP, but is W[$t$]-hard\u0000for every $t$. In contrast to the fact that OLP parameterized by the height lies in XP, it\u0000turns out that CLP is NP-hard even when restricted to instances of height 4. We\u0000complement this result by showing that CLP can be solved in polynomial time for\u0000instances of height at most 3.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201018","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}

引用次数: 0

Fréchet Edit Distance 弗雷谢特编辑距离

arXiv - CS - Computational Geometry Pub Date : 2024-03-19 DOI: arxiv-2403.12878

Emily Fox, Amir Nayyeri, Jonathan James Perry, Benjamin Raichel

引用次数: 0

Semi-Algebraic Off-line Range Searching and Biclique Partitions in the Plane 半代数离线范围搜索和平面上的双斜分区

arXiv - CS - Computational Geometry Pub Date : 2024-03-18 DOI: arxiv-2403.12276

Pankaj K. Agarwal, Esther Ezra, Micha Sharir

{"title":"Semi-Algebraic Off-line Range Searching and Biclique Partitions in the Plane","authors":"Pankaj K. Agarwal, Esther Ezra, Micha Sharir","doi":"arxiv-2403.12276","DOIUrl":"https://doi.org/arxiv-2403.12276","url":null,"abstract":"Let $P$ be a set of $m$ points in ${mathbb R}^2$, let $Sigma$ be a set of\u0000$n$ semi-algebraic sets of constant complexity in ${mathbb R}^2$, let $(S,+)$\u0000be a semigroup, and let $w: P rightarrow S$ be a weight function on the points\u0000of $P$. We describe a randomized algorithm for computing $w(Pcapsigma)$ for\u0000every $sigmainSigma$ in overall expected time $O^*bigl(\u0000m^{frac{2s}{5s-4}}n^{frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n bigr)$,\u0000where $s>0$ is a constant that bounds the maximum complexity of the regions of\u0000$Sigma$, and where the $O^*(cdot)$ notation hides subpolynomial factors. For\u0000$sge 3$, surprisingly, this bound is smaller than the best-known bound for\u0000answering $m$ such queries in an on-line manner. The latter takes\u0000$O^*(m^{frac{s}{2s-1}}n^{frac{2s-2}{2s-1}}+m+n)$ time. Let $Phi: Sigma times P rightarrow {0,1}$ be the Boolean predicate (of\u0000constant complexity) such that $Phi(sigma,p) = 1$ if $pinsigma$ and $0$\u0000otherwise, and let $Sigmamathop{Phi} P = { (sigma,p) in Sigmatimes P\u0000mid Phi(sigma,p)=1}$. Our algorithm actually computes a partition\u0000${mathcal B}_Phi$ of $Sigmamathop{Phi} P$ into bipartite cliques\u0000(bicliques) of size (i.e., sum of the sizes of the vertex sets of its\u0000bicliques) $O^*bigl( m^{frac{2s}{5s-4}}n^{frac{5s-6}{5s-4}} + m^{2/3}n^{2/3}\u0000+ m + n bigr)$. It is straightforward to compute $w(Pcapsigma)$ for all\u0000$sigmain Sigma$ from ${mathcal B}_Phi$. Similarly, if $eta: Sigma\u0000rightarrow S$ is a weight function on the regions of $Sigma$,\u0000$sum_{sigmain Sigma: p in sigma} eta(sigma)$, for every point $pin P$,\u0000can be computed from ${mathcal B}_Phi$ in a straightforward manner. A recent\u0000work of Chan et al. solves the online version of this dual point enclosure\u0000problem within the same performance bound as our off-line solution. We also\u0000mention a few other applications of computing ${mathcal B}_Phi$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165877","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}

引用次数: 0

Robustly Guarding Polygons 稳健地保护多边形

arXiv - CS - Computational Geometry Pub Date : 2024-03-18 DOI: arxiv-2403.11861

Rathish Das, Omrit Filtser, Matthew J. Katz, Joseph S. B. Mitchell

引用次数: 0

Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane 半代数平面范围刺击、射线射击和交点计数

arXiv - CS - Computational Geometry Pub Date : 2024-03-18 DOI: arxiv-2403.12303

Timothy M. Chan, Pingan Cheng, Da Wei Zheng

{"title":"Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane","authors":"Timothy M. Chan, Pingan Cheng, Da Wei Zheng","doi":"arxiv-2403.12303","DOIUrl":"https://doi.org/arxiv-2403.12303","url":null,"abstract":"Polynomial partitioning techniques have recently led to improved geometric\u0000data structures for a variety of fundamental problems related to semialgebraic\u0000range searching and intersection searching in 3D and higher dimensions (e.g.,\u0000see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021;\u0000Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to\u0000improved algorithms for offline versions of semialgebraic range searching in\u00002D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that\u0000these techniques can yield new data structures for a number of other 2D\u0000problems even for online queries: 1. Semialgebraic range stabbing. We present a data structure for $n$\u0000semialgebraic ranges in 2D of constant description complexity with\u0000$O(n^{3/2+varepsilon})$ preprocessing time and space, so that we can count the\u0000number of ranges containing a query point in $O(n^{1/4+varepsilon})$ time, for\u0000an arbitrarily small constant $varepsilon>0$. 2. Ray shooting amid algebraic arcs. We present a data structure for $n$\u0000algebraic arcs in 2D of constant description complexity with\u0000$O(n^{3/2+varepsilon})$ preprocessing time and space, so that we can find the\u0000first arc hit by a query (straight-line) ray in $O(n^{1/4+varepsilon})$ time. 3. Intersection counting amid algebraic arcs. We present a data structure for\u0000$n$ algebraic arcs in 2D of constant description complexity with\u0000$O(n^{3/2+varepsilon})$ preprocessing time and space, so that we can count the\u0000number of intersection points with a query algebraic arc of constant\u0000description complexity in $O(n^{1/2+varepsilon})$ time. In particular, this\u0000implies an $O(n^{3/2+varepsilon})$-time algorithm for counting intersections\u0000between two sets of $n$ algebraic arcs in 2D.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166128","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}

引用次数: 0

Ipelets for the Convex Polygonal Geometry 凸多边形几何的插梢

arXiv - CS - Computational Geometry Pub Date : 2024-03-15 DOI: arxiv-2403.10033

Nithin Parepally, Ainesh Chatterjee, Auguste Gezalyan, Hongyang Du, Sukrit Mangla, Kenny Wu, Sarah Hwang, David Mount

引用次数: 0

Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing 基于布局图变换的不等边圆打包的解-散列搜索

arXiv - CS - Computational Geometry Pub Date : 2024-03-10 DOI: arxiv-2403.06211

Jianrong Zhou, Jiyao He, Kun He

{"title":"Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing","authors":"Jianrong Zhou, Jiyao He, Kun He","doi":"arxiv-2403.06211","DOIUrl":"https://doi.org/arxiv-2403.06211","url":null,"abstract":"The problem of packing unequal circles into a circular container stands as a\u0000classic and challenging optimization problem in computational geometry. This\u0000study introduces a suite of innovative and efficient methods to tackle this\u0000problem. Firstly, we present a novel layout-graph transformation method that\u0000represents configurations as graphs, together with an inexact hash method\u0000facilitating fast comparison of configurations for isomorphism or similarity.\u0000Leveraging these advancements, we propose an Iterative Solution-Hashing Search\u0000algorithm adept at circumventing redundant exploration through efficient\u0000configuration recording. Additionally, we introduce several enhancements to\u0000refine the optimization and search processes, including an adaptive adjacency\u0000maintenance method, an efficient vacancy detection technique, and a\u0000Voronoi-based locating method. Through comprehensive computational experiments\u0000across various benchmark instances, our algorithm demonstrates superior\u0000performance over existing state-of-the-art methods, showcasing remarkable\u0000applicability and versatility. Notably, our algorithm surpasses the best-known\u0000results for 56 out of 179 benchmark instances while achieving parity with the\u0000remaining instances.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106160","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}

引用次数: 0

A Clique-Based Separator for Intersection Graphs of Geodesic Disks in $mathbb{R}^2$ $mathbb{R}^2$中大地圆盘相交图的基于簇的分离器

arXiv - CS - Computational Geometry Pub Date : 2024-03-07 DOI: arxiv-2403.04905

Boris Aronov, Mark de Berg, Leonidas Theocharous

{"title":"A Clique-Based Separator for Intersection Graphs of Geodesic Disks in $mathbb{R}^2$","authors":"Boris Aronov, Mark de Berg, Leonidas Theocharous","doi":"arxiv-2403.04905","DOIUrl":"https://doi.org/arxiv-2403.04905","url":null,"abstract":"Let $d$ be a (well-behaved) shortest-path metric defined on a path-connected\u0000subset of $mathbb{R}^2$ and let $mathcal{D}={D_1,ldots,D_n}$ be a set of\u0000geodesic disks with respect to the metric $d$. We prove that\u0000$mathcal{G}^{times}(mathcal{D})$, the intersection graph of the disks in\u0000$mathcal{D}$, has a clique-based separator consisting of\u0000$O(n^{3/4+varepsilon})$ cliques. This significantly extends the class of\u0000objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for $q$-COLORING that runs in\u0000time $2^{O(n^{3/4+varepsilon})}$, assuming the boundaries of the disks $D_i$\u0000can be computed in polynomial time. We also use our clique-based separator to\u0000obtain a simple, efficient, and almost exact distance oracle for intersection\u0000graphs of geodesic disks. Our distance oracle uses $O(n^{7/4+varepsilon})$\u0000storage and can report the hop distance between any two nodes in\u0000$mathcal{G}^{times}(mathcal{D})$ in $O(n^{3/4+varepsilon})$ time, up to an\u0000additive error of one. So far, distance oracles with an additive error of one\u0000that use subquadratic storage and sublinear query time were not known for such\u0000general graph classes.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097727","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}

引用次数: 0

Fine-Grained Complexity of Earth Mover's Distance under Translation 平移条件下地球移动距离的精细复杂性

arXiv - CS - Computational Geometry Pub Date : 2024-03-07 DOI: arxiv-2403.04356

Karl Bringmann, Frank Staals, Karol Węgrzycki, Geert van Wordragen

引用次数: 0

A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon 简单多边形中近似最远邻查询的核心集

arXiv - CS - Computational Geometry Pub Date : 2024-03-07 DOI: arxiv-2403.04513

Mark de Berg, Leonidas Theocharous

引用次数: 0