arXiv - CS - Computational Geometry最新文献_第3页

Ironing the Graphs: Toward a Correct Geometric Analysis of Large-Scale Graphs 熨平图形：实现大规模图形的正确几何分析

arXiv - CS - Computational Geometry Pub Date : 2024-07-31 DOI: arxiv-2407.21609

Saloua Naama, Kavé Salamatian, Francesco Bronzino

{"title":"Ironing the Graphs: Toward a Correct Geometric Analysis of Large-Scale Graphs","authors":"Saloua Naama, Kavé Salamatian, Francesco Bronzino","doi":"arxiv-2407.21609","DOIUrl":"https://doi.org/arxiv-2407.21609","url":null,"abstract":"Graph embedding approaches attempt to project graphs into geometric entities,\u0000i.e, manifolds. The idea is that the geometric properties of the projected\u0000manifolds are helpful in the inference of graph properties. However, if the\u0000choice of the embedding manifold is incorrectly performed, it can lead to\u0000incorrect geometric inference. In this paper, we argue that the classical\u0000embedding techniques cannot lead to correct geometric interpretation as they\u0000miss the curvature at each point, of manifold. We advocate that for doing\u0000correct geometric interpretation the embedding of graph should be done over\u0000regular constant curvature manifolds. To this end, we present an embedding\u0000approach, the discrete Ricci flow graph embedding (dRfge) based on the discrete\u0000Ricci flow that adapts the distance between nodes in a graph so that the graph\u0000can be embedded onto a constant curvature manifold that is homogeneous and\u0000isotropic, i.e., all directions are equivalent and distances comparable,\u0000resulting in correct geometric interpretations. A major contribution of this\u0000paper is that for the first time, we prove the convergence of discrete Ricci\u0000flow to a constant curvature and stable distance metrics over the edges. A\u0000drawback of using the discrete Ricci flow is the high computational complexity\u0000that prevented its usage in large-scale graph analysis. Another contribution of\u0000this paper is a new algorithmic solution that makes it feasible to calculate\u0000the Ricci flow for graphs of up to 50k nodes, and beyond. The intuitions behind\u0000the discrete Ricci flow make it possible to obtain new insights into the\u0000structure of large-scale graphs. We demonstrate this through a case study on\u0000analyzing the internet connectivity structure between countries at the BGP\u0000level.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870280","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

Map-Matching Queries under Fréchet Distance on Low-Density Spanners 低密度跨度弗雷谢特距离下的地图匹配查询

arXiv - CS - Computational Geometry Pub Date : 2024-07-27 DOI: arxiv-2407.19304

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Aleksandr Popov, Sampson Wong

引用次数: 0

Carving Polytopes with Saws in 3D 用锯子雕刻 3D 多面体

arXiv - CS - Computational Geometry Pub Date : 2024-07-22 DOI: arxiv-2407.15981

Eliot W. Robson, Jack Spalding-Jamieson, Da Wei Zheng

引用次数: 0

SimpleSets: Capturing Categorical Point Patterns with Simple Shapes SimpleSets：用简单形状捕捉分类点模式

arXiv - CS - Computational Geometry Pub Date : 2024-07-19 DOI: arxiv-2407.14433

Steven van den Broek, Wouter Meulemans, Bettina Speckmann

{"title":"SimpleSets: Capturing Categorical Point Patterns with Simple Shapes","authors":"Steven van den Broek, Wouter Meulemans, Bettina Speckmann","doi":"arxiv-2407.14433","DOIUrl":"https://doi.org/arxiv-2407.14433","url":null,"abstract":"Points of interest on a map such as restaurants, hotels, or subway stations,\u0000give rise to categorical point data: data that have a fixed location and one or\u0000more categorical attributes. Consequently, recent years have seen various set\u0000visualization approaches that visually connect points of the same category to\u0000support users in understanding the spatial distribution of categories. Existing\u0000methods use complex and often highly irregular shapes to connect points of the\u0000same category, leading to high cognitive load for the user. In this paper we\u0000introduce SimpleSets, which uses simple shapes to enclose categorical point\u0000patterns, thereby providing a clean overview of the data distribution.\u0000SimpleSets is designed to visualize sets of points with a single categorical\u0000attribute; as a result, the point patterns enclosed by SimpleSets form a\u0000partition of the data. We give formal definitions of point patterns that\u0000correspond to simple shapes and describe an algorithm that partitions\u0000categorical points into few such patterns. Our second contribution is a\u0000rendering algorithm that transforms a given partition into a clean set of\u0000shapes resulting in an aesthetically pleasing set visualization. Our algorithm\u0000pays particular attention to resolving intersections between nearby shapes in a\u0000consistent manner. We compare SimpleSets to the state-of-the-art set\u0000visualizations using standard datasets from the literature.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743995","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

Complexity of 2D Snake Cube Puzzles 二维蛇形立方体谜题的复杂性

arXiv - CS - Computational Geometry Pub Date : 2024-07-14 DOI: arxiv-2407.10323

MIT Hardness Group, Nithid Anchaleenukoon, Alex Dang, Erik D. Demaine, Kaylee Ji, Pitchayut Saengrungkongka

引用次数: 0

Synchronous Diffusion for Unsupervised Smooth Non-Rigid 3D Shape Matching 用于无监督平滑非刚性三维形状匹配的同步扩散技术

arXiv - CS - Computational Geometry Pub Date : 2024-07-11 DOI: arxiv-2407.08244

Dongliang Cao, Zorah Laehner, Florian Bernard

引用次数: 0

Approximating the Fréchet distance when only one curve is $c$-packed 当只有一条曲线上有 $c$ 包络时的近似弗雷谢特距离

arXiv - CS - Computational Geometry Pub Date : 2024-07-06 DOI: arxiv-2407.05114

Joachim Gudmundsson, Michael Mai, Sampson Wong

引用次数: 0

Bicriteria approximation for minimum dilation graph augmentation 最小扩张图扩展的双标准近似法

arXiv - CS - Computational Geometry Pub Date : 2024-07-05 DOI: arxiv-2407.04614

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Sampson Wong

引用次数: 0

The Fréchet Distance Unleashed: Approximating a Dog with a Frog 释放弗雷谢特距离：用青蛙逼近狗

arXiv - CS - Computational Geometry Pub Date : 2024-07-03 DOI: arxiv-2407.03101

Sariel Har-Peled, Benjamin Raichel, Eliot W. Robson

{"title":"The Fréchet Distance Unleashed: Approximating a Dog with a Frog","authors":"Sariel Har-Peled, Benjamin Raichel, Eliot W. Robson","doi":"arxiv-2407.03101","DOIUrl":"https://doi.org/arxiv-2407.03101","url":null,"abstract":"We show that a minor variant of the continuous Fr'echet distance between\u0000polygonal curves can be computed using essentially the same algorithm used to\u0000solve the discrete version, thus dramatically simplifying the algorithm for\u0000computing it. The new variant is not necessarily monotone, but this shortcoming\u0000can be easily handled via refinement. Combined with a Dijkstra/Prim type algorithm, this leads to a realization of\u0000the Fr'echet distance (i.e., a morphing) that is locally optimal (aka locally\u0000correct), that is both easy to compute, and in practice, takes near linear time\u0000on many inputs. The new morphing has the property that the leash is always as\u0000short-as-possible. We implemented the new algorithm, and developed various strategies to get a\u0000fast execution in practice. Among our new contributions is a new simplification\u0000strategy that is distance-sensitive, and enables us to compute the exact\u0000continuous Fr'echet distance in near linear time in practice. We preformed\u0000extensive experiments on our new algorithm, and released texttt{Julia} and\u0000texttt{Python} packages with these new implementations.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551373","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

Efficient Exact Algorithms for Minimum Covering of Orthogonal Polygons with Squares 用正方形最小覆盖正交多边形的高效精确算法

arXiv - CS - Computational Geometry Pub Date : 2024-07-02 DOI: arxiv-2407.02658

Anubhav Dhar, Subham Ghosh, Sudeshna Kolay

{"title":"Efficient Exact Algorithms for Minimum Covering of Orthogonal Polygons with Squares","authors":"Anubhav Dhar, Subham Ghosh, Sudeshna Kolay","doi":"arxiv-2407.02658","DOIUrl":"https://doi.org/arxiv-2407.02658","url":null,"abstract":"The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input an\u0000orthogonal polygon $P$ without holes with $n$ vertices, where vertices have\u0000integral coordinates. The aim is to find a minimum number of axis-parallel,\u0000possibly overlapping squares which lie completely inside $P$, such that their\u0000union covers the entire region inside $P$. Aupperle et.\u0000al~cite{aupperle1988covering} provide an $mathcal O(N^{1.5})$-time algorithm\u0000to solve OPCS for orthogonal polygons without holes, where $N$ is the number of\u0000integral lattice points lying in the interior or on the boundary of $P$.\u0000Designing algorithms for OPCS with a running time polynomial in $n$ (the number\u0000of vertices of $P$) was discussed as an open question in\u0000cite{aupperle1988covering}, since $N$ can be exponentially larger than $n$. In\u0000this paper we design a polynomial-time exact algorithm for OPCS with a running\u0000time of $mathcal O(n^{14})$. We also consider the following structural parameterized version of the\u0000problem. A knob in an orthogonal polygon is a polygon edge whose both endpoints\u0000are convex polygon vertices. Given an input orthogonal polygon with $n$\u0000vertices and $k$ knobs, we design an algorithm for OPCS with running time\u0000$mathcal O(n^2 + k^{14} cdot n)$. In cite{aupperle1988covering}, the Orthogonal Polygon with Holes Covering\u0000with Squares (OPCSH) problem is also studied where orthogonal polygon could\u0000have holes, and the objective is to find a minimum square covering of the input\u0000polygon. This is shown to be NP-complete. We think there is an error in the\u0000existing proof in cite{aupperle1988covering}, where a reduction from Planar\u00003-CNF is shown. We fix this error in the proof with an alternate construction\u0000of one of the gadgets used in the reduction, hence completing the proof of\u0000NP-completeness of OPCSH.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551374","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