A Clique-Based Separator for Intersection Graphs of Geodesic Disks in \(\mathbb {R}^2\)

IF 0.7 4区 计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING
Boris Aronov, Mark de Berg, Leonidas Theocharous
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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.

一种基于团的测地线盘交点图分割方法 \(\mathbb {R}^2\)
设d是在\(\mathbb {R}^2\)的路径连通子集上定义的(表现良好的)最短路径度量,设\(\mathcal {D}=\{D_1,\ldots,D_n\}\)是一组关于度量d的测地线磁盘。我们证明\(\mathcal {D}\)中磁盘的相交图\(\mathcal {G}^{\times }(\mathcal {D})\)有一个由\(O(n^{3/4+\varepsilon })\)块组成的基于团的分隔符。这极大地扩展了交集图具有小的基于团的分隔符的对象类。我们基于团的分隔符产生q-Coloring算法,该算法运行时间为\(2^{O(n^{3/4+\varepsilon })}\),假设磁盘的边界\(D_i\)可以在多项式时间内计算。我们还使用我们的基于团的分隔符来获得一个简单、高效、几乎精确的测地线盘相交图的距离oracle。我们的距离oracle使用\(O(n^{7/4+\varepsilon })\)存储,并且可以在\(O(n^{3/4+\varepsilon })\)时间内报告\(\mathcal {G}^{\times }(\mathcal {D})\)中任意两个节点之间的跳距离,最多可添加误差为1。到目前为止,对于这种一般的图类,使用次二次存储和次线性查询时间的加性误差为1的距离预言器还不为人所知。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Algorithmica
Algorithmica 工程技术-计算机:软件工程
CiteScore
2.80
自引率
9.10%
发文量
158
审稿时长
12 months
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.
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