2026-07-21T23:49:09Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23412 2025-10-02T12:54:27Z ddc:004 open_access

Near-Optimal Directed Low-Diameter Decompositions Bringmann, Karl Fischer, Nick Haeupler, Bernhard Latypov, Rustam Low Diameter Decompositions Expander Decompositions Directed Graphs Low Diameter Decompositions (LDDs) are invaluable tools in the design of combinatorial graph algorithms. While historically they have been applied mainly to undirected graphs, in the recent breakthrough for the negative-length Single Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen [FOCS '22] extended the use of LDDs to directed graphs for the first time. Specifically, their LDD deletes each edge with probability at most O(1/D ⋅ log²n), while ensuring that each strongly connected component in the remaining graph has a (weak) diameter of at most D. In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of O(1/D ⋅ log n log log n). This improves the previous bound by nearly a logarithmic factor and closely approaches the lower bound of Ω(1/D ⋅ log n). With significantly more technical effort, we also develop two efficient algorithms for computing our LDDs: a deterministic algorithm that runs in time Õ(m poly(D)) and a randomized algorithm that runs in near-linear time Õ(m). We believe that our work provides a solid conceptual and technical foundation for future research relying on directed LDDs, which will undoubtedly follow soon. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Karl Bringmann and Nick Fischer and Bernhard Haeupler and Rustam Latypov 2025 Is Part Of LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2025.35 urn:nbn:de:0030-drops-234125 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.35 eng https://creativecommons.org/licenses/by/4.0/legalcode