2026-07-21T14:16:36Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11696 2024-03-06T10:48:28Z ddc:004 open_access

Algorithms and Lower Bounds for Cycles and Walks: Small Space and Sparse Graphs Lincoln, Andrea Vyas, Nikhil k-cycle Space Savitch Sparse Graphs Max-3-SAT 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-ε)) time for some ε>0 then, a 2^(n(1-ε')) time algorithm exists for Max-3-SAT for some ε'>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 Savitch’s 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^Ω(lg(n)) time, we show that undirected 2k-cycle in O(lg(n)) space requires n^Ω(lg(k)) time. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andrea Lincoln and Nikhil Vyas 2020 Is Part Of LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2020.11 urn:nbn:de:0030-drops-116969 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.11 eng https://creativecommons.org/licenses/by/3.0/legalcode