arXiv - CS - Computational Geometry最新文献

Evolving Distributions Under Local Motion 局部运动下的分布演变

arXiv - CS - Computational Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11779

Aditya Acharya, David M. Mount

{"title":"Evolving Distributions Under Local Motion","authors":"Aditya Acharya, David M. Mount","doi":"arxiv-2409.11779","DOIUrl":"https://doi.org/arxiv-2409.11779","url":null,"abstract":"Geometric data sets arising in modern applications are often very large and\u0000change dynamically over time. A popular framework for dealing with such data\u0000sets is the evolving data framework, where a discrete structure continuously\u0000varies over time due to the unseen actions of an evolver, which makes small\u0000changes to the data. An algorithm probes the current state through an oracle,\u0000and the objective is to maintain a hypothesis of the data set's current state\u0000that is close to its actual state at all times. In this paper, we apply this\u0000framework to maintaining a set of $n$ point objects in motion in\u0000$d$-dimensional Euclidean space. To model the uncertainty in the object\u0000locations, both the ground truth and hypothesis are based on spatial\u0000probability distributions, and the distance between them is measured by the\u0000Kullback-Leibler divergence (relative entropy). We introduce a simple and\u0000intuitive motion model where with each time step, the distance that any object\u0000can move is a fraction of the distance to its nearest neighbor. We present an\u0000algorithm that, in steady state, guarantees a distance of $O(n)$ between the\u0000true and hypothesized placements. We also show that for any algorithm in this\u0000model, there is an evolver that can generate a distance of $Omega(n)$,\u0000implying that our algorithm is asymptotically optimal.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251452","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

Minimum Plane Bichromatic Spanning Trees 最小平面双色生成树

arXiv - CS - Computational Geometry Pub Date : 2024-09-18 DOI: arxiv-2409.11614

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, Csaba D. Tóth

{"title":"Minimum Plane Bichromatic Spanning Trees","authors":"Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, Csaba D. Tóth","doi":"arxiv-2409.11614","DOIUrl":"https://doi.org/arxiv-2409.11614","url":null,"abstract":"For a set of red and blue points in the plane, a minimum bichromatic spanning\u0000tree (MinBST) is a shortest spanning tree of the points such that every edge\u0000has a red and a blue endpoint. A MinBST can be computed in $O(nlog n)$ time\u0000where $n$ is the number of points. In contrast to the standard Euclidean MST,\u0000which is always plane (noncrossing), a MinBST may have edges that cross each\u0000other. However, we prove that a MinBST is quasi-plane, that is, it does not\u0000contain three pairwise crossing edges, and we determine the maximum number of\u0000crossings. Moreover, we study the problem of finding a minimum plane bichromatic\u0000spanning tree (MinPBST) which is a shortest bichromatic spanning tree with\u0000pairwise noncrossing edges. This problem is known to be NP-hard. The previous\u0000best approximation algorithm, due to Borgelt et al. (2009), has a ratio of\u0000$O(sqrt{n})$. It is also known that the optimum solution can be computed in\u0000polynomial time in some special cases, for instance, when the points are in\u0000convex position, collinear, semi-collinear, or when one color class has\u0000constant size. We present an $O(log n)$-factor approximation algorithm for the\u0000general case.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251451","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

New Lower Bound and Algorithms for Online Geometric Hitting Set Problem 在线几何命中集问题的新下限和算法

arXiv - CS - Computational Geometry Pub Date : 2024-09-17 DOI: arxiv-2409.11166

Minati De, Ratnadip Mandal, Satyam Singh

{"title":"New Lower Bound and Algorithms for Online Geometric Hitting Set Problem","authors":"Minati De, Ratnadip Mandal, Satyam Singh","doi":"arxiv-2409.11166","DOIUrl":"https://doi.org/arxiv-2409.11166","url":null,"abstract":"The hitting set problem is one of the fundamental problems in combinatorial\u0000optimization and is well-studied in offline setup. We consider the online\u0000hitting set problem, where only the set of points is known in advance, and\u0000objects are introduced one by one. Our objective is to maintain a minimum-sized\u0000hitting set by making irrevocable decisions. Here, we present the study of two\u0000variants of the online hitting set problem depending on the point set. In the\u0000first variant, we consider the point set to be the entire $mathbb{Z}^d$, while\u0000in the second variant, we consider the point set to be a finite subset of\u0000$mathbb{R}^2$. For hitting similarly sized {$alpha$-fat objects} in $mathbb{R}^d$ with\u0000diameters in the range $[1, M]$ using points in $mathbb{Z}^d$, we propose a\u0000deterministic algorithm having a competitive ratio of at most\u0000${lfloorfrac{2}{alpha}+2rfloor^d}$\u0000$left(lfloorlog_{2}Mrfloor+1right)$. This improves the current best-known\u0000upper bound due to Alefkhani et al. [WAOA'23]. Then, for homothetic hypercubes\u0000in $mathbb{R}^d$ with side lengths in the range $[1, M]$ using points in\u0000$mathbb{Z}^d$, we propose a randomized algorithm having a competitive ratio of\u0000$O(d^2log M)$. To complement this result, we show that no randomized algorithm\u0000can have a competitive ratio better than $Omega(dlog M)$. This improves the\u0000current best-known (deterministic) upper and lower bound of $25^dlog M$ and\u0000$Omega(log M)$, respectively, due to Alefkhani et al. [WAOA'23]. Next, we consider the hitting set problem when the point set consists of $n$\u0000points in $mathbb{R}^2$ and the objects are homothetic regular $k$-gons having\u0000diameter in the range $[1, M]$. We present an $O(log nlog M)$ competitive\u0000randomized algorithm. In particular, for a fixed $M$ this result partially\u0000answers an open question for squares proposed by Khan et al. [SoCG'23] and\u0000Alefkhani et al. [WAOA'23].","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251453","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

Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing 使用位置敏感哈希算法对合并树进行快速比较分析

arXiv - CS - Computational Geometry Pub Date : 2024-09-13 DOI: arxiv-2409.08519

Weiran Lyu, Raghavendra Sridharamurthy, Jeff M. Phillips, Bei Wang

{"title":"Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing","authors":"Weiran Lyu, Raghavendra Sridharamurthy, Jeff M. Phillips, Bei Wang","doi":"arxiv-2409.08519","DOIUrl":"https://doi.org/arxiv-2409.08519","url":null,"abstract":"Scalar field comparison is a fundamental task in scientific visualization. In\u0000topological data analysis, we compare topological descriptors of scalar fields\u0000-- such as persistence diagrams and merge trees -- because they provide\u0000succinct and robust abstract representations. Several similarity measures for\u0000topological descriptors seem to be both asymptotically and practically\u0000efficient with polynomial time algorithms, but they do not scale well when\u0000handling large-scale, time-varying scientific data and ensembles. In this\u0000paper, we propose a new framework to facilitate the comparative analysis of\u0000merge trees, inspired by tools from locality sensitive hashing (LSH). LSH\u0000hashes similar objects into the same hash buckets with high probability. We\u0000propose two new similarity measures for merge trees that can be computed via\u0000LSH, using new extensions to Recursive MinHash and subpath signature,\u0000respectively. Our similarity measures are extremely efficient to compute and\u0000closely resemble the results of existing measures such as merge tree edit\u0000distance or geometric interleaving distance. Our experiments demonstrate the\u0000utility of our LSH framework in applications such as shape matching,\u0000clustering, key event detection, and ensemble summarization.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251455","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

Computing shortest paths amid non-overlapping weighted disks 计算非重叠加权磁盘中的最短路径

arXiv - CS - Computational Geometry Pub Date : 2024-09-13 DOI: arxiv-2409.08869

Prosenjit Bose, Jean-Lou De Carufel, Guillermo Esteban, Anil Maheshwari

引用次数: 0

Revisiting Accurate Geometry for Morse-Smale Complexes 重新审视莫尔斯-马勒复合物的精确几何学

arXiv - CS - Computational Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.05532

Son Le Thanh, Michael Ankele, Tino Weinkauf

{"title":"Revisiting Accurate Geometry for Morse-Smale Complexes","authors":"Son Le Thanh, Michael Ankele, Tino Weinkauf","doi":"arxiv-2409.05532","DOIUrl":"https://doi.org/arxiv-2409.05532","url":null,"abstract":"The Morse-Smale complex is a standard tool in visual data analysis. The\u0000classic definition is based on a continuous view of the gradient of a scalar\u0000function where its zeros are the critical points. These points are connected\u0000via gradient curves and surfaces emanating from saddle points, known as\u0000separatrices. In a discrete setting, the Morse-Smale complex is commonly\u0000extracted by constructing a combinatorial gradient assuming the steepest\u0000descent direction. Previous works have shown that this method results in a\u0000geometric embedding of the separatrices that can be fundamentally different\u0000from those in the continuous case. To achieve a similar embedding, different\u0000approaches for constructing a combinatorial gradient were proposed. In this\u0000paper, we show that these approaches generate a different topology, i.e., the\u0000connectivity between critical points changes. Additionally, we demonstrate that\u0000the steepest descent method can compute topologically and geometrically\u0000accurate Morse-Smale complexes when applied to certain types of grids. Based on\u0000these observations, we suggest a method to attain both geometric and\u0000topological accuracy for the Morse-Smale complex of data sampled on a uniform\u0000grid.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214996","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

Harmonic Chain Barcode and Stability 谐波链条码和稳定性

arXiv - CS - Computational Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.06093

Salman Parsa, Bei Wang

引用次数: 0

HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees HyperSteiner：计算启发式双曲斯坦纳最小树

arXiv - CS - Computational Geometry Pub Date : 2024-09-09 DOI: arxiv-2409.05671

Alejandro García-Castellanos, Aniss Aiman Medbouhi, Giovanni Luca Marchetti, Erik J. Bekkers, Danica Kragic

引用次数: 0

COVID19-CBABM: A City-Based Agent Based Disease Spread Modeling Framework COVID19-CBABM：基于城市代理的疾病传播建模框架

arXiv - CS - Computational Geometry Pub Date : 2024-09-08 DOI: arxiv-2409.05235

Raunak Sarbajna, Karima Elgarroussi, Hoang D Vo, Jianyuan Ni, Christoph F. Eick

{"title":"COVID19-CBABM: A City-Based Agent Based Disease Spread Modeling Framework","authors":"Raunak Sarbajna, Karima Elgarroussi, Hoang D Vo, Jianyuan Ni, Christoph F. Eick","doi":"arxiv-2409.05235","DOIUrl":"https://doi.org/arxiv-2409.05235","url":null,"abstract":"In response to the ongoing pandemic and health emergency of COVID-19, several\u0000models have been used to understand the dynamics of virus spread. Some employ\u0000mathematical models like the compartmental SEIHRD approach and others rely on\u0000agent-based modeling (ABM). In this paper, a new city-based agent-based\u0000modeling approach called COVID19-CBABM is introduced. It considers not only the\u0000transmission mechanism simulated by the SEHIRD compartments but also models\u0000people movements and their interactions with their surroundings, particularly\u0000their interactions at different types of Points of Interest (POI), such as\u0000supermarkets. Through the development of knowledge extraction procedures for\u0000Safegraph data, our approach simulates realistic conditions based on spatial\u0000patterns and infection conditions considering locations where people spend\u0000their time in a given city. Our model was implemented in Python using the\u0000Mesa-Geo framework. COVID19-CBABM is portable and can be easily extended by\u0000adding more complicated scenarios. Therefore, it is a useful tool to assist the\u0000government and health authorities in evaluating strategic decisions and actions\u0000efficiently against this epidemic, using the unique mobility patterns of each\u0000city.","PeriodicalId":501570,"journal":{"name":"arXiv - CS - Computational Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142214997","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

Morphing Planar Graph Drawings via Orthogonal Box Drawings 通过正交框图变形平面图形绘制

arXiv - CS - Computational Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04074

Therese Biedl, Anna Lubiw, Jack Spalding-Jamieson

引用次数: 0