2026-07-23T03:47:44Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16427 2024-03-06T10:57:27Z ddc:004 open_access

Near-Optimal Decremental Hopsets with Applications Łącki, Jakub Nazari, Yasamin Dynamic Algorithms Data Structures Shortest Paths Hopsets Given a weighted undirected graph G = (V,E,w), a hopset H of hopbound β and stretch (1+ε) is a set of edges such that for any pair of nodes u, v ∈ V, there is a path in G ∪ H of at most β hops, whose length is within a (1+ε) factor from the distance between u and v in G. We show the first efficient decremental algorithm for maintaining hopsets with a polylogarithmic hopbound. The update time of our algorithm matches the best known static algorithm up to polylogarithmic factors. All the previous decremental hopset constructions had a superpolylogarithmic (but subpolynomial) hopbound of 2^{log^{Ω(1)} n} [Bernstein, FOCS'09; HKN, FOCS'14; Chechik, FOCS'18]. By applying our decremental hopset construction, we get improved or near optimal bounds for several distance problems. Most importantly, we show how to decrementally maintain (2k-1)(1+ε)-approximate all-pairs shortest paths (for any constant k ≥ 2), in Õ(n^{1/k}) amortized update time and O(k) query time. This improves (by a polynomial factor) over the update-time of the best previously known decremental algorithm in the constant query time regime. Moreover, it improves over the result of [Chechik, FOCS'18] that has a query time of O(log log(nW)), where W is the aspect ratio, and the amortized update time is n^{1/k}⋅(1/ε)^{Õ(√{log n})}). For sparse graphs our construction nearly matches the best known static running time / query time tradeoff. We also obtain near-optimal bounds for maintaining approximate multi-source shortest paths and distance sketches, and get improved bounds for approximate single-source shortest paths. Our algorithms are randomized and our bounds hold with high probability against an oblivious adversary. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jakub Łącki and Yasamin Nazari 2022 Is Part Of LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2022.86 urn:nbn:de:0030-drops-164277 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.86 eng https://creativecommons.org/licenses/by/4.0/legalcode