Deterministic decremental single source shortest paths: beyond the o(mn) bound

A. Bernstein, S. Chechik
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引用次数: 58

Abstract

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of Õ(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.
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确定性递减单源最短路径:超过0 (mn)界限
本文考虑递减单源最短路径(SSSP)问题,其中给定一个图G和一个源节点s,目标是在一系列在线对抗性边删除下保持s与G中所有其他节点之间的最短路径。在他们的开创性工作中,Even和Shiloach [JACM 1981]提出了一个精确的解决方案,所有边缘删除的总更新时间仅为O(mn)。他们的经典算法是三十年来最著名的递减SSSP问题的结果,即使是在允许近似最短路径的情况下。Bernstein和Roditty [SODA 2011]对eveno - shiloach算法进行了第一次改进,对于无权无向图,他们提出了一种近似(1+)算法,查询时间恒定,总更新时间为O(n2+O(1/√logn)))。这项工作引发了一系列新的结果,最近Henzinger, Krinninger和Nanongkai的突破达到高潮[FOCS 14],他们提出了一个-近似算法,其总更新时间接近线性O(m1+ O(1/√logn)))。在这篇论文中,他们提出了一个重大的开放性问题,即结果的非随机化问题。事实上,所有已知的Even-Shiloach算法的改进都是随机的。所有这些算法都从一个小的节点子集中维护一些截断的最短路径树。虽然在随机设置中,可以对对手“隐藏”这些节点,但在确定性设置中,这是不可能的:对手可以删除与这些节点接触的所有边,从而迫使算法选择一组新的节点,并引发最短路径的新计算。在本文中,我们提出了第一个确定性递减SSSP算法,该算法打破了O(mn)总更新时间的Even-Shiloach界,用于无权无向图。我们的算法近似为(1 + n),总更新时间为Õ(n2)。我们的算法也可以在增量设置中实现相同的边界。值得一提的是,对于m = Ω(n2−1/√log(n))的密集实例,我们的算法也比所有现有的随机化算法快。
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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