We study the dual of Philo's shortest line segment problem which asks to find the optimal line segments passing through two given points, with a common endpoint, and with the other endpoints on a given line. The provided solution uses multivariable calculus and geometry methods. Interesting connections with the angle bisector of the triangle are explored.
{"title":"The dual of Philo's shortest line segment problem","authors":"Yagub N. Aliyev","doi":"arxiv-2406.05702","DOIUrl":"https://doi.org/arxiv-2406.05702","url":null,"abstract":"We study the dual of Philo's shortest line segment problem which asks to find\u0000the optimal line segments passing through two given points, with a common\u0000endpoint, and with the other endpoints on a given line. The provided solution\u0000uses multivariable calculus and geometry methods. Interesting connections with\u0000the angle bisector of the triangle are explored.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515265","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}
Sándor P. Fekete, Joseph S. B. Mitchell, Christian Rieck, Christian Scheffer, Christiane Schmidt
We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon $mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at least $1$, and a rational number $ell$; decide whether there is a set of vertex guards such that $mathcal{P}$ is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least $ell$. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2$. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least $2$. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most $2$.
{"title":"Dispersive Vertex Guarding for Simple and Non-Simple Polygons","authors":"Sándor P. Fekete, Joseph S. B. Mitchell, Christian Rieck, Christian Scheffer, Christiane Schmidt","doi":"arxiv-2406.05861","DOIUrl":"https://doi.org/arxiv-2406.05861","url":null,"abstract":"We study the Dispersive Art Gallery Problem with vertex guards: Given a\u0000polygon $mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at\u0000least $1$, and a rational number $ell$; decide whether there is a set of\u0000vertex guards such that $mathcal{P}$ is guarded, and the minimum geodesic\u0000Euclidean distance between any two guards (the so-called dispersion distance)\u0000is at least $ell$. We show that it is NP-complete to decide whether a polygon with holes has a\u0000set of vertex guards with dispersion distance $2$. On the other hand, we\u0000provide an algorithm that places vertex guards in simple polygons at dispersion\u0000distance at least $2$. This result is tight, as there are simple polygons in\u0000which any vertex guard set has a dispersion distance of at most $2$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509233","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}
Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, Morteza Saghafian
Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry.
{"title":"Chromatic Topological Data Analysis","authors":"Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert Edelsbrunner, Morteza Saghafian","doi":"arxiv-2406.04102","DOIUrl":"https://doi.org/arxiv-2406.04102","url":null,"abstract":"Exploring the shape of point configurations has been a key driver in the\u0000evolution of TDA (short for topological data analysis) since its infancy. This\u0000survey illustrates the recent efforts to broaden these ideas to model spatial\u0000interactions among multiple configurations, each distinguished by a color. It\u0000describes advances in this area and prepares the ground for further exploration\u0000by mentioning unresolved questions and promising research avenues while\u0000focusing on the overlap with discrete geometry.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553300","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}
The well-known textsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the textsc{$k$-Watchmen Routes} problem, in a simple polygon $P$. We look at both the version in which the $k$ watchmen must collectively see all of $P$, and the quota version in which they must see a predetermined fraction of $P$'s area. We give an exact pseudopolynomial time algorithm for the textsc{$k$-Watchmen Routes} problem in a simple orthogonal polygon $P$ with the constraint that watchmen must move on axis-parallel segments, and there is a given common starting point on the boundary. Further, we give a fully polynomial-time approximation scheme and a constant-factor approximation for unconstrained movement. For the quota version, we give a constant-factor approximation in a simple polygon, utilizing the solution to the (single) textsc{Quota Watchman Route} problem.
{"title":"Multirobot Watchman Routes in a Simple Polygon","authors":"Joseph S. B. Mitchell, Linh Nguyen","doi":"arxiv-2405.21034","DOIUrl":"https://doi.org/arxiv-2405.21034","url":null,"abstract":"The well-known textsc{Watchman Route} problem seeks a shortest route in a\u0000polygonal domain from which every point of the domain can be seen. In this\u0000paper, we study the cooperative variant of the problem, namely the\u0000textsc{$k$-Watchmen Routes} problem, in a simple polygon $P$. We look at both\u0000the version in which the $k$ watchmen must collectively see all of $P$, and the\u0000quota version in which they must see a predetermined fraction of $P$'s area. We give an exact pseudopolynomial time algorithm for the textsc{$k$-Watchmen\u0000Routes} problem in a simple orthogonal polygon $P$ with the constraint that\u0000watchmen must move on axis-parallel segments, and there is a given common\u0000starting point on the boundary. Further, we give a fully polynomial-time\u0000approximation scheme and a constant-factor approximation for unconstrained\u0000movement. For the quota version, we give a constant-factor approximation in a\u0000simple polygon, utilizing the solution to the (single) textsc{Quota Watchman\u0000Route} problem.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254167","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}
In this work, we study the parallel complexity of the Euclidean minimum-weight perfect matching (EWPM) problem. Here our graph is the complete bipartite graph $G$ on two sets of points $A$ and $B$ in $mathbb{R}^2$ and the weight of each edge is the Euclidean distance between the corresponding points. The weighted perfect matching problem on general bipartite graphs is known to be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner, Gurjar, and Thierauf, 2016]. Both of these results work only when the weights are of $O(log n)$ bits. It is a long-standing open question to show the problem to be in NC. First, we show that for EWPM, a linear number of bits of approximation is required to distinguish between the minimum-weight perfect matching and other perfect matchings. Next, we show that the EWPM problem that allows up to $frac{1}{poly(n)}$ additive error, is in NC.
{"title":"Geometric Bipartite Matching is in NC","authors":"Sujoy Bhore, Sarfaraz Equbal, Rohit Gurjar","doi":"arxiv-2405.18833","DOIUrl":"https://doi.org/arxiv-2405.18833","url":null,"abstract":"In this work, we study the parallel complexity of the Euclidean\u0000minimum-weight perfect matching (EWPM) problem. Here our graph is the complete\u0000bipartite graph $G$ on two sets of points $A$ and $B$ in $mathbb{R}^2$ and the\u0000weight of each edge is the Euclidean distance between the corresponding points.\u0000The weighted perfect matching problem on general bipartite graphs is known to\u0000be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner,\u0000Gurjar, and Thierauf, 2016]. Both of these results work only when the weights\u0000are of $O(log n)$ bits. It is a long-standing open question to show the\u0000problem to be in NC. First, we show that for EWPM, a linear number of bits of approximation is\u0000required to distinguish between the minimum-weight perfect matching and other\u0000perfect matchings. Next, we show that the EWPM problem that allows up to\u0000$frac{1}{poly(n)}$ additive error, is in NC.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141198360","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}
$ newcommand{cardin}[1]{left| {#1} right|}% newcommand{Graph}{Mh{mathsf{G}}}% providecommand{G}{Graph}% renewcommand{G}{Graph}% providecommand{GA}{Mh{H}}% renewcommand{GA}{Mh{H}}% newcommand{VV}{Mh{mathsf{V}}}% newcommand{VX}[1]{VVpth{#1}}% providecommand{EE}{Mh{mathsf{E}}}% renewcommand{EE}{Mh{mathsf{E}}}% newcommand{Re}{mathbb{R}} newcommand{reals}{mathbb{R}} newcommand{SetX}{mathsf{X}} newcommand{rad}{r} newcommand{Mh}[1]{#1} newcommand{query}{q} newcommand{eps}{varepsilon} newcommand{VorX}[1]{mathcal{V} pth{#1}} newcommand{Polygon}{mathsf{P}} newcommand{IntRange}[1]{[ #1 ]} newcommand{Space}{overline{mathsf{m}}} newcommand{pth}[2][!]{#1left({#2}right)} newcommand{polylog}{mathrm{polylog}} newcommand{N}{mathbb N} newcommand{Z}{mathbb Z} newcommand{pt}{p} newcommand{distY}[2]{left| {#1} - {#2} right|} newcommand{ptq}{q} newcommand{pts}{s}$ For an undirected graph $mathsf{G}=(mathsf{V}, mathsf{E})$, with $n$ vertices and $m$ edges, the emph{densest subgraph} problem, is to compute a subset $S subseteq mathsf{V}$ which maximizes the ratio $|mathsf{E}_S| / |S|$, where $mathsf{E}_S subseteq mathsf{E}$ is the set of all edges of $mathsf{G}$ with endpoints in $S$. The densest subgraph problem is a well studied problem in computer science. Existing exact and approximation algorithms for computing the densest subgraph require $Omega(m)$ time. We present near-linear time (in $n$) approximation algorithms for the densest subgraph problem on emph{implicit} geometric intersection graphs, where the vertices are explicitly given but not the edges. As a concrete example, we consider $n$ disks in the plane with arbitrary radii and present two different approximation algorithms.
{"title":"Approximating Densest Subgraph in Geometric Intersection Graphs","authors":"Sariel Har-Peled, Rahul Saladi","doi":"arxiv-2405.18337","DOIUrl":"https://doi.org/arxiv-2405.18337","url":null,"abstract":"$ newcommand{cardin}[1]{left| {#1} right|}%\u0000newcommand{Graph}{Mh{mathsf{G}}}% providecommand{G}{Graph}%\u0000renewcommand{G}{Graph}% providecommand{GA}{Mh{H}}%\u0000renewcommand{GA}{Mh{H}}% newcommand{VV}{Mh{mathsf{V}}}%\u0000newcommand{VX}[1]{VVpth{#1}}% providecommand{EE}{Mh{mathsf{E}}}%\u0000renewcommand{EE}{Mh{mathsf{E}}}% newcommand{Re}{mathbb{R}}\u0000newcommand{reals}{mathbb{R}} newcommand{SetX}{mathsf{X}}\u0000newcommand{rad}{r} newcommand{Mh}[1]{#1} newcommand{query}{q}\u0000newcommand{eps}{varepsilon} newcommand{VorX}[1]{mathcal{V} pth{#1}}\u0000newcommand{Polygon}{mathsf{P}} newcommand{IntRange}[1]{[ #1 ]}\u0000newcommand{Space}{overline{mathsf{m}}}\u0000newcommand{pth}[2][!]{#1left({#2}right)}\u0000newcommand{polylog}{mathrm{polylog}} newcommand{N}{mathbb N}\u0000newcommand{Z}{mathbb Z} newcommand{pt}{p} newcommand{distY}[2]{left|\u0000{#1} - {#2} right|} newcommand{ptq}{q} newcommand{pts}{s}$ For an\u0000undirected graph $mathsf{G}=(mathsf{V}, mathsf{E})$, with $n$ vertices and\u0000$m$ edges, the emph{densest subgraph} problem, is to compute a subset $S\u0000subseteq mathsf{V}$ which maximizes the ratio $|mathsf{E}_S| / |S|$, where\u0000$mathsf{E}_S subseteq mathsf{E}$ is the set of all edges of $mathsf{G}$\u0000with endpoints in $S$. The densest subgraph problem is a well studied problem\u0000in computer science. Existing exact and approximation algorithms for computing\u0000the densest subgraph require $Omega(m)$ time. We present near-linear time (in\u0000$n$) approximation algorithms for the densest subgraph problem on\u0000emph{implicit} geometric intersection graphs, where the vertices are\u0000explicitly given but not the edges. As a concrete example, we consider $n$\u0000disks in the plane with arbitrary radii and present two different approximation\u0000algorithms.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165755","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}
In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c_1, c_2,..., c_{alpha}}. We take a subset S of V, such that for every vertex v in VS, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such a S as a consistent subset (CS) The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum size. It is established that MCS is NP-complete for general graphs, including planar graphs. We expand our study to interval graphs and circle graphs in an attempt to gain a complete understanding of the computational complexity of the MCS problem across various graph classes. The strict consistent subset is a variant of consistent subset problems. We take a subset S^{prime} of V, such that for every vertex v in VS^{prime}, all the vertices in its set of nearest neighbors in S have the same color as v. We refer to such a S^{prime} as a strict consistent subset (SCS). The Minimum Strict Consistent Subset (MSCS) problem is the computation of a consistent subset of the minimum size. We demonstrate that MSCS is NP-hard in general graphs. We show a 2-approximation in trees. Later, we show polynomial-time algorithms in trees. Later, we demonstrate faster polynomial-time algorithms in paths, spiders, and combs.
在连通的简单图 G = (V,E)中，V 的每个顶点都由颜色集合 C={c_1, c_2,..., c_{alpha}} 中的一种颜色着色。我们取 V 的一个子集 S，这样对于 VS 中的每个顶点 v，至少有一个相同颜色的顶点出现在它在 S 中的近邻集合中。 我们把这样的 S 称为一致子集（CS）。最小一致子集（MCS）问题是计算一个最小大小的一致子集。对于一般图（包括平面图）来说，MCS 是一个 NP-完全问题。我们将研究扩展到区间图和圆图，试图全面了解不同图类中 MCS 问题的计算复杂性。严格一致子集是一致子集问题的一个变种。我们取 V 的一个子集 S^{prime}，对于 VS^{prime} 中的每个顶点 v，S 中其最近邻集合中的所有顶点都与 v 具有相同的颜色，我们把这样的 S^{prime} 称为严格一致子集（SCS）。最小严格一致子集（MSCS）问题就是计算一个最小大小的一致子集。我们证明，在一般图中，MSCS 是 NP 难问题。我们展示了树中的 2 近似值。随后，我们展示了树中的多项式时间算法。随后，我们展示了路径、蜘蛛和梳状图中更快的多项式时间算法。
{"title":"Minimum Strict Consistent Subset in Paths, Spiders, Combs and Trees","authors":"Bubai Manna","doi":"arxiv-2405.18569","DOIUrl":"https://doi.org/arxiv-2405.18569","url":null,"abstract":"In a connected simple graph G = (V,E), each vertex of V is colored by a color\u0000from the set of colors C={c_1, c_2,..., c_{alpha}}. We take a subset S of V,\u0000such that for every vertex v in VS, at least one vertex of the same color is\u0000present in its set of nearest neighbors in S. We refer to such a S as a\u0000consistent subset (CS) The Minimum Consistent Subset (MCS) problem is the\u0000computation of a consistent subset of the minimum size. It is established that\u0000MCS is NP-complete for general graphs, including planar graphs. We expand our\u0000study to interval graphs and circle graphs in an attempt to gain a complete\u0000understanding of the computational complexity of the MCS problem across various\u0000graph classes. The strict consistent subset is a variant of consistent subset\u0000problems. We take a subset S^{prime} of V, such that for every vertex v in\u0000VS^{prime}, all the vertices in its set of nearest neighbors in S have the\u0000same color as v. We refer to such a S^{prime} as a strict consistent subset\u0000(SCS). The Minimum Strict Consistent Subset (MSCS) problem is the computation\u0000of a consistent subset of the minimum size. We demonstrate that MSCS is NP-hard in general graphs. We show a\u00002-approximation in trees. Later, we show polynomial-time algorithms in trees.\u0000Later, we demonstrate faster polynomial-time algorithms in paths, spiders, and\u0000combs.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141192342","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}
Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid
Let $H$ be an edge-weighted graph, and let $G$ be a subgraph of $H$. We say that $G$ is an $f$-fault-tolerant $t$-spanner for $H$, if the following is true for any subset $F$ of at most $f$ edges of $G$: For any two vertices $p$ and $q$, the shortest-path distance between $p$ and $q$ in the graph $G setminus F$ is at most $t$ times the shortest-path distance between $p$ and $q$ in the graph $H setminus F$. Recently, Bodwin, Haeupler, and Parter generalized this notion to the case when $F$ can be any set of edges in $G$, as long as the maximum degree of $F$ is at most $f$. They gave constructions for general graphs $H$. We first consider the case when $H$ is a complete graph whose vertex set is an arbitrary metric space. We show that if this metric space contains a $t$-spanner with $m$ edges, then it also contains a graph $G$ with $O(fm)$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor $O(ft)$. Next, we consider the case when $H$ is a complete graph whose vertex set is a metric space that admits a well-separated pair decomposition. We show that, if the metric space has such a decomposition of size $m$, then it contains a graph with at most $(2f+1)^2 m$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor at most $1+varepsilon$, for any given $varepsilon > 0$. For example, if the vertex set is a set of $n$ points in $mathbb{R}^d$ ($d$ being a constant) or a set of $n$ points in a metric space of bounded doubling dimension, then the spanner has $O(f^2 n)$ edges. Finally, for the case when $H$ is a complete graph on $n$ points in $mathbb{R}^d$, we show how natural variants of the Yao- and $Theta$-graphs lead to graphs with $O(fn)$ edges, that are resilient to edge faults of maximum degree $f$ and have stretch factor at most $1+varepsilon$, for any given $varepsilon > 0$.
{"title":"Metric and Geometric Spanners that are Resilient to Degree-Bounded Edge Faults","authors":"Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid","doi":"arxiv-2405.18134","DOIUrl":"https://doi.org/arxiv-2405.18134","url":null,"abstract":"Let $H$ be an edge-weighted graph, and let $G$ be a subgraph of $H$. We say\u0000that $G$ is an $f$-fault-tolerant $t$-spanner for $H$, if the following is true\u0000for any subset $F$ of at most $f$ edges of $G$: For any two vertices $p$ and\u0000$q$, the shortest-path distance between $p$ and $q$ in the graph $G setminus\u0000F$ is at most $t$ times the shortest-path distance between $p$ and $q$ in the\u0000graph $H setminus F$. Recently, Bodwin, Haeupler, and Parter generalized this notion to the case\u0000when $F$ can be any set of edges in $G$, as long as the maximum degree of $F$\u0000is at most $f$. They gave constructions for general graphs $H$. We first consider the case when $H$ is a complete graph whose vertex set is\u0000an arbitrary metric space. We show that if this metric space contains a\u0000$t$-spanner with $m$ edges, then it also contains a graph $G$ with $O(fm)$\u0000edges, that is resilient to edge faults of maximum degree $f$ and has stretch\u0000factor $O(ft)$. Next, we consider the case when $H$ is a complete graph whose vertex set is a\u0000metric space that admits a well-separated pair decomposition. We show that, if\u0000the metric space has such a decomposition of size $m$, then it contains a graph\u0000with at most $(2f+1)^2 m$ edges, that is resilient to edge faults of maximum\u0000degree $f$ and has stretch factor at most $1+varepsilon$, for any given\u0000$varepsilon > 0$. For example, if the vertex set is a set of $n$ points in\u0000$mathbb{R}^d$ ($d$ being a constant) or a set of $n$ points in a metric space\u0000of bounded doubling dimension, then the spanner has $O(f^2 n)$ edges. Finally, for the case when $H$ is a complete graph on $n$ points in\u0000$mathbb{R}^d$, we show how natural variants of the Yao- and $Theta$-graphs\u0000lead to graphs with $O(fn)$ edges, that are resilient to edge faults of maximum\u0000degree $f$ and have stretch factor at most $1+varepsilon$, for any given\u0000$varepsilon > 0$.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165662","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}
Lattice structures have been widely used in applications due to their superior mechanical properties. To fabricate such structures, a geometric processing step called triangulation is often employed to transform them into the STL format before sending them to 3D printers. Because lattice structures tend to have high geometric complexity, this step usually generates a large amount of triangles, a memory and compute-intensive task. This problem manifests itself clearly through large-scale lattice structures that have millions or billions of struts. To address this problem, this paper proposes to transform a lattice structure into an intermediate model called meta-mesh before undergoing real triangulation. Compared to triangular meshes, meta-meshes are very lightweight and much less compute-demanding. The meta-mesh can also work as a base mesh reusable for conveniently and efficiently triangulating lattice structures with arbitrary resolutions. A CPU+GPU asynchronous meta-meshing pipeline has been developed to efficiently generate meta-meshes from lattice structures. It shifts from the thread-centric GPU algorithm design paradigm commonly used in CAD to the recent warp-centric design paradigm to achieve high performance. This is achieved by a new data compression method, a GPU cache-aware data structure, and a workload-balanced scheduling method that can significantly reduce memory divergence and branch divergence. Experimenting with various billion-scale lattice structures, the proposed method is seen to be two orders of magnitude faster than previously achievable.
{"title":"Meta-meshing and triangulating lattice structures at a large scale","authors":"Qiang Zou, Yunzhu Gao, Guoyue Luo, Sifan Chen","doi":"arxiv-2405.15197","DOIUrl":"https://doi.org/arxiv-2405.15197","url":null,"abstract":"Lattice structures have been widely used in applications due to their\u0000superior mechanical properties. To fabricate such structures, a geometric\u0000processing step called triangulation is often employed to transform them into\u0000the STL format before sending them to 3D printers. Because lattice structures\u0000tend to have high geometric complexity, this step usually generates a large\u0000amount of triangles, a memory and compute-intensive task. This problem\u0000manifests itself clearly through large-scale lattice structures that have\u0000millions or billions of struts. To address this problem, this paper proposes to\u0000transform a lattice structure into an intermediate model called meta-mesh\u0000before undergoing real triangulation. Compared to triangular meshes,\u0000meta-meshes are very lightweight and much less compute-demanding. The meta-mesh\u0000can also work as a base mesh reusable for conveniently and efficiently\u0000triangulating lattice structures with arbitrary resolutions. A CPU+GPU\u0000asynchronous meta-meshing pipeline has been developed to efficiently generate\u0000meta-meshes from lattice structures. It shifts from the thread-centric GPU\u0000algorithm design paradigm commonly used in CAD to the recent warp-centric\u0000design paradigm to achieve high performance. This is achieved by a new data\u0000compression method, a GPU cache-aware data structure, and a workload-balanced\u0000scheduling method that can significantly reduce memory divergence and branch\u0000divergence. Experimenting with various billion-scale lattice structures, the\u0000proposed method is seen to be two orders of magnitude faster than previously\u0000achievable.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165845","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}
In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c1, c2,..., c_{alpha}}$. We take a subset S of V, such that for every vertex v in VS, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such a S as a consistent subset. The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum size. It is established that MCS is NP-complete for general graphs, including planar graphs. We expand our study to interval graphs and circle graphs in an attempt to gain a complete understanding of the computational complexity of the mcs problem across various graph classes. This work introduces an (4alpha+ 2)- approximation algorithm for MCS in interval graphs where alpha is the number of colors in the interval graphs. Later, we show that in circle graphs, MCS is APX-hard.
在连通的简单图 G = (V,E)中，V 的每个顶点都由颜色集合 C={c1, c2,..., c_{alpha}}$ 中的一种颜色着色。我们取 V 的一个子集 S，对于 VS 中的每个顶点 v，至少有一个相同颜色的顶点出现在它在 S 中的近邻集合中。最小一致子集（MCS）问题就是计算最小大小的一致子集。对于一般图（包括平面图）来说，MCS 是一个 NP-完全问题。我们将研究扩展到区间图和圆图，试图全面了解各种图类的（MCS）问题的计算复杂性。这项工作介绍了区间图中 MCS 的 (4alpha+ 2)- 近似算法，其中 alpha 是区间图中颜色的数量。
{"title":"Minimum Consistent Subset in Interval Graphs and Circle Graphs","authors":"Bubai Manna","doi":"arxiv-2405.14493","DOIUrl":"https://doi.org/arxiv-2405.14493","url":null,"abstract":"In a connected simple graph G = (V,E), each vertex of V is colored by a color\u0000from the set of colors C={c1, c2,..., c_{alpha}}$. We take a subset S of V,\u0000such that for every vertex v in VS, at least one vertex of the same color is\u0000present in its set of nearest neighbors in S. We refer to such a S as a\u0000consistent subset. The Minimum Consistent Subset (MCS) problem is the\u0000computation of a consistent subset of the minimum size. It is established that\u0000MCS is NP-complete for general graphs, including planar graphs. We expand our\u0000study to interval graphs and circle graphs in an attempt to gain a complete\u0000understanding of the computational complexity of the mcs problem across\u0000various graph classes. This work introduces an (4alpha+ 2)- approximation algorithm for MCS in\u0000interval graphs where alpha is the number of colors in the interval graphs.\u0000Later, we show that in circle graphs, MCS is APX-hard.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141148947","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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