文章基本信息

标题：Algorithms and Lower Bounds for Cycles and Walks: Small Space and Sparse Graphs
本地全文：下载
作者：Andrea Lincoln ; Nikhil Vyas
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：151
页码：1-17
DOI：10.4230/LIPIcs.ITCS.2020.11
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We consider space-efficient algorithms and conditional time lower bounds for finding cycles and walks in graphs. We give a reduction that connects the running time of undirected 2k-cycle to finding directed odd cycles, s-t connectivity in directed graphs, and Max-3-SAT. For example, we show that if 2k-cycle on O(n)-edge graphs can be solved in O(n^(1.5-Îµ)) time for some Îµ>0 then, a 2^(n(1-Îµ')) time algorithm exists for Max-3-SAT for some Îµ'>0. Additionally, we give a tight combinatorial lower bound for 2k-cycle detection, specifically when k is odd, of m^{2k/(k+1) +o(1)} given the Combinatorial k-Clique Hypothesis. On the algorithms side, we present a randomized algorithm for directed s-t connectivity using O(lg(n)^2) space and O(n^{lg(n)/2 + o(lg(n))}) expected time, giving a time improvement over Savitchâs famous algorithm, which takes at least n^{lg(n) - o(lg(n))} time. Under the conjecture that every O(lg(n)^2)-space algorithm for directed s-t connectivity requires n^Î©(lg(n)) time, we show that undirected 2k-cycle in O(lg(n)) space requires n^Î©(lg(k)) time.
关键词：k-cycle; Space; Savitch; Sparse Graphs; Max-3-SAT