{"title":"A Clique-Based Separator for Intersection Graphs of Geodesic Disks in \\(\\mathbb {R}^2\\)","authors":"Boris Aronov, Mark de Berg, Leonidas Theocharous","doi":"10.1007/s00453-025-01337-5","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>d</i> be a (well-behaved) shortest-path metric defined on a path-connected subset of <span>\\(\\mathbb {R}^2\\)</span> and let <span>\\(\\mathcal {D}=\\{D_1,\\ldots,D_n\\}\\)</span> be a set of geodesic disks with respect to the metric <i>d</i>. We prove that <span>\\(\\mathcal {G}^{\\times }(\\mathcal {D})\\)</span>, the intersection graph of the disks in <span>\\(\\mathcal {D}\\)</span>, has a clique-based separator consisting of <span>\\(O(n^{3/4+\\varepsilon })\\)</span> cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for <i>q</i>-<span>Coloring</span> that runs in time <span>\\(2^{O(n^{3/4+\\varepsilon })}\\)</span>, assuming the boundaries of the disks <span>\\(D_i\\)</span> can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses <span>\\(O(n^{7/4+\\varepsilon })\\)</span> storage and can report the hop distance between any two nodes in <span>\\(\\mathcal {G}^{\\times }(\\mathcal {D})\\)</span> in <span>\\(O(n^{3/4+\\varepsilon })\\)</span> time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 12","pages":"1997 - 2017"},"PeriodicalIF":0.7000,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-025-01337-5.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithmica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00453-025-01337-5","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of \(\mathbb {R}^2\) and let \(\mathcal {D}=\{D_1,\ldots,D_n\}\) be a set of geodesic disks with respect to the metric d. We prove that \(\mathcal {G}^{\times }(\mathcal {D})\), the intersection graph of the disks in \(\mathcal {D}\), has a clique-based separator consisting of \(O(n^{3/4+\varepsilon })\) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time \(2^{O(n^{3/4+\varepsilon })}\), assuming the boundaries of the disks \(D_i\) can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses \(O(n^{7/4+\varepsilon })\) storage and can report the hop distance between any two nodes in \(\mathcal {G}^{\times }(\mathcal {D})\) in \(O(n^{3/4+\varepsilon })\) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.
期刊介绍:
Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential.
Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming.
In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.