2 research outputs found

    Fast Generation of Random Spanning Trees and the Effective Resistance Metric

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    We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected O~(m4/3)\tilde{O}(m^{4/3}) time. This improves over the best previously known bound of min(O~(mn),O(nω))\min(\tilde{O}(m\sqrt{n}),O(n^{\omega})) -- that follows from the work of Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph

    Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

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    In the decremental (1+ϵ)(1+\epsilon)-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph G=(V,E)G=(V,E) with n=V,m=En = |V|, m = |E|, undergoing edge deletions, and a distinguished source sVs \in V, and we are asked to process edge deletions efficiently and answer queries for distance estimates dist~G(s,v)\widetilde{\mathbf{dist}}_G(s,v) for each vVv \in V, at any stage, such that distG(s,v)dist~G(s,v)(1+ϵ)distG(s,v)\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ \epsilon)\mathbf{dist}_G(s,v). In the decremental (1+ϵ)(1+\epsilon)-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates dist~G(u,v)\widetilde{\mathbf{dist}}_G(u,v) for every u,vVu,v \in V. In this article, we consider the problems for undirected, unweighted graphs. We present a new \emph{deterministic} algorithm for the decremental (1+ϵ)(1+\epsilon)-approximate SSSP problem that takes total update time O(mn0.5+o(1))O(mn^{0.5 + o(1)}). Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time O~(n2)\tilde{O}(n^2) and the best existing algorithm for sparse graphs with running time O~(n1.25m)\tilde{O}(n^{1.25}\sqrt{m}) [SODA 2017] whenever m=O(n1.5o(1))m = O(n^{1.5 - o(1)}). In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic \emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental (1+ϵ)(1+\epsilon)-approximate APSP problem with near-optimal total running time O~(mn/ϵ)\tilde{O}(mn /\epsilon) matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].Comment: Appeared in SODA'2
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