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Faster Dynamic 2-Edge Connectivity in Directed Graphs

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Titel: Faster Dynamic 2-Edge Connectivity in Directed Graphs
Autoren: Georgiadis, Loukas, Giannis, Konstantinos, Italiano, Giuseppe F.
Weitere Verfasser: Loukas Georgiadis and Konstantinos Giannis and Giuseppe F. Italiano
Verlagsinformationen: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2025.
Publikationsjahr: 2025
Schlagwörter: Connectivity, dynamic algorithms, ddc:004, directed graphs
Beschreibung: Let G be a directed graph with n vertices and m edges. We present a deterministic algorithm that maintains the 2-edge-connected components of G under a sequence of m edge insertions, with a total running time of O(n² log n). This significantly improves upon the previous best bound of O(mn) for graphs that are not very sparse. After each insertion, our algorithm supports the following queries with asymptotically optimal efficiency: - Test in constant time whether two query vertices v and w are 2-edge-connected in G. - Report in O(n) time all the 2-edge-connected components of G. Our approach builds on the recent framework of Georgiadis, Italiano, and Kosinas [FOCS 2024] for computing the 3-edge-connected components of a directed graph in linear time, which leverages the minset-poset technique of Gabow [TALG 2016]. Additionally, we provide a deterministic decremental algorithm for maintaining 2-edge-connectivity in strongly connected directed graphs. Given a sequence of m edge deletions, our algorithm maintains the 2-edge-connected components in total time n^(2+o(1)), while supporting the same queries as the incremental algorithm. This result assumes that the edges of a fixed spanning tree of G and of its reverse graph G^R are not deleted. Previously, the best known bound for the decremental problem was O(mn log n), obtained by a randomized algorithm without restrictions on the deletions. In contrast to prior dynamic algorithms for 2-edge-connectivity in directed graphs, our method avoids the incremental computation of dominator trees, thereby circumventing the known conditional lower bound of Ω(mn).
Publikationsart: Conference object
Dateibeschreibung: application/pdf
Sprache: English
DOI: 10.4230/lipics.esa.2025.26
Zugangs-URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.26
Rights: CC BY
Dokumentencode: edsair.od......1814..a36aaabc65a844bb3fd7477d5ed0e270
Datenbank: OpenAIRE
Beschreibung
Abstract:Let G be a directed graph with n vertices and m edges. We present a deterministic algorithm that maintains the 2-edge-connected components of G under a sequence of m edge insertions, with a total running time of O(n² log n). This significantly improves upon the previous best bound of O(mn) for graphs that are not very sparse. After each insertion, our algorithm supports the following queries with asymptotically optimal efficiency: - Test in constant time whether two query vertices v and w are 2-edge-connected in G. - Report in O(n) time all the 2-edge-connected components of G. Our approach builds on the recent framework of Georgiadis, Italiano, and Kosinas [FOCS 2024] for computing the 3-edge-connected components of a directed graph in linear time, which leverages the minset-poset technique of Gabow [TALG 2016]. Additionally, we provide a deterministic decremental algorithm for maintaining 2-edge-connectivity in strongly connected directed graphs. Given a sequence of m edge deletions, our algorithm maintains the 2-edge-connected components in total time n^(2+o(1)), while supporting the same queries as the incremental algorithm. This result assumes that the edges of a fixed spanning tree of G and of its reverse graph G^R are not deleted. Previously, the best known bound for the decremental problem was O(mn log n), obtained by a randomized algorithm without restrictions on the deletions. In contrast to prior dynamic algorithms for 2-edge-connectivity in directed graphs, our method avoids the incremental computation of dominator trees, thereby circumventing the known conditional lower bound of Ω(mn).
DOI:10.4230/lipics.esa.2025.26