Online Spanners in Metric Spaces

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-03-12 DOI:10.1137/22m1534572

Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, Csaba D. Tóth

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1030-1056, March 2024.
Abstract. Given a metric space [math], a weighted graph [math] over [math] is a metric [math]-spanner of [math] if for every [math], [math], where [math] is the shortest path metric in [math]. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points [math], where the points are presented one at a time (i.e., after [math] steps, we see [math]). The algorithm is allowed to add edges to the spanner when a new point arrives; however, it is not allowed to remove any edge from the spanner. The goal is to maintain a [math]-spanner [math] for [math] for all [math], while minimizing the number of edges, and their total weight. We construct online [math]-spanners in the Euclidean [math]-space, [math]-spanners for general metrics, and [math]-spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a [math]-spanner with competitive ratio [math], bypassing the classic lower bound [math] for lightness, which compares the weight of the spanner to that of the minimum spanning tree.

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公制空间中的在线施展器

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 1030-1056 页，2024 年 3 月。摘要。给定一个度量空间[math]，如果对于每一个[math]，[math]，其中[math]是[math]中的最短路径度量，则[math]上的加权图[math]是[math]的度量[math]-spanner。在本文中，我们将在在线设置中为度量空间中的有限集构建spanner。在这里，我们给定了一个点序列 [math]，点是一个一个呈现的（即在 [math] 步之后，我们看到 [math]）。当有新的点出现时，算法可以在[spanner]中添加边；但不允许从[spanner]中移除任何边。我们的目标是为[math]的所有[math]保持一个[math]-spanner[math]，同时最大限度地减少边的数量及其总权重。我们构建了欧氏[math]空间的在线[math]扫描器、一般度量的[math]扫描器和超度量的[math]扫描器。最值得注意的是，在欧几里得平面上，我们构建了一个具有竞争比[math]的[math]扫描器，绕过了经典的轻度下限[math]，它将扫描器的权重与最小生成树的权重进行了比较。

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

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.