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Near-Optimal Algorithms for Reachability, Strongly-Connected Components and Shortest Paths in Partially Dynamic Digraphs
arXiv - CS - Data Structures and Algorithms Pub Date : 2020-11-27 , DOI: arxiv-2011.13702
Maximilian Probst Gutenberg

In this thesis, we present new techniques to deal with fundamental algorithmic graph problems where graphs are directed and partially dynamic, i.e. undergo either a sequence of edge insertions or deletions: - Single-Source Reachability (SSR), - Strongly-Connected Components (SCCs), and - Single-Source Shortest Paths (SSSP). These problems have recently received an extraordinary amount of attention due to their role as subproblems in various more complex and notoriously hard graph problems, especially to compute flows, bipartite matchings and cuts. Our techniques lead to the first near-optimal data structures for these problems in various different settings. Letting $n$ denote the number of vertices in the graph and by $m$ the maximum number of edges in any version of the graph, we obtain - the first randomized data structure to maintain SSR and SCCs in near-optimal total update time $\tilde{O}(m)$ in a graph undergoing edge deletions. - the first randomized data structure to maintain SSSP in partially dynamic graphs in total update time $\tilde{O}(n^2)$ which is near-optimal in dense graphs. - the first deterministic data structures for SSR and SCC for graphs undergoing edge deletions, and for SSSP in partially dynamic graphs that improve upon the $O(mn)$ total update time by Even and Shiloach from 1981 that is often considered to be a fundamental barrier.

中文翻译:

部分动态有向图的可达性,强连接组件和最短路径的近似最佳算法

在本文中,我们提出了新的技术来处理基本的算法图问题,在这些问题中,图是有向图和部分动态图,即经历一系列边插入或删除:-单源可达性(SSR),-强连接组件(SCC) ),以及-单源最短路径(SSSP)。这些问题作为子问题在各种更复杂且众所周知的硬图问题中的作用最近受到了极大的关注,尤其是在计算流量,二分匹配和割裂方面。我们的技术导致在各种不同设置下针对这些问题的第一个近乎最优的数据结构。假设$ n $表示图形中的顶点数,而$ m $表示图形的任何版本中的最大边数,我们获得了-第一个随机数据结构,用于在发生边缘删除的图中以接近最优的总更新时间$ \ tilde {O}(m)$维持SSR和SCC。-第一个在总更新时间$ \ tilde {O}(n ^ 2)$中保持部分动态图中的SSSP的随机数据结构,在密集图中接近最佳。-用于SSR和SCC的第一个确定性数据结构,用于经过边缘删除的图形,以及用于SSSP的部分动态图形,该结构改善了Even和Shiloach自1981年以来的总更新时间$ O(mn)$,这通常被认为是基本的屏障。
更新日期:2020-12-01
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