We consider the distributed and parallel construction of low-diameter decompositions with strong diameter. We present algorithms for arbitrary undirected, weighted graphs and also for undirected, weighted graphs that can be separated through k ∈ Õ(1) shortest paths. This class of graphs includes planar graphs, graphs of bounded treewidth, and graphs that exclude a fixed minor K_r. Our algorithms work in the PRAM, CONGEST, and the novel HYBRID communication model and are competitive in all relevant parameters. Given 𝒟 > 0, our low-diameter decomposition algorithm divides the graph into connected clusters of strong diameter 𝒟. For an arbitrary graph, an edge e ∈ E of length 𝓁_e is cut between two clusters with probability O(𝓁_e⋅log(n)/𝒟). If the graph can be separated by k ∈ Õ(1) paths, the probability improves to O(𝓁_e⋅log(log n)/𝒟). In either case, the decompositions can be computed in Õ(1) depth and Õ(m) work in the PRAM and Õ(1) time in the HYBRID model. In CONGEST, the runtimes are Õ(HD + √n) and Õ(HD) respectively. All these results hold w.h.p. Broadly speaking, we present distributed and parallel implementations of sequential divide-and-conquer algorithms where we replace exact shortest paths with approximate shortest paths. In contrast to exact paths, these can be efficiently computed in the distributed and parallel setting [STOC '22]. Further, and perhaps more importantly, we show that instead of explicitly computing vertex-separators to enable efficient parallelization of these algorithms, it suffices to sample a few random paths of bounded length and the nodes close to them. Thereby, we do not require complex embeddings whose implementation is unknown in the distributed and parallel setting.
@InProceedings{dou_et_al:LIPIcs.ITCS.2025.45, author = {Dou, Jinfeng and G\"{o}tte, Thorsten and Hillebrandt, Henning and Scheideler, Christian and Werthmann, Julian}, title = {{Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs}}, booktitle = {16th Innovations in Theoretical Computer Science Conference (ITCS 2025)}, pages = {45:1--45:26}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-361-4}, ISSN = {1868-8969}, year = {2025}, volume = {325}, editor = {Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.45}, URN = {urn:nbn:de:0030-drops-226734}, doi = {10.4230/LIPIcs.ITCS.2025.45}, annote = {Keywords: Distributed Graph Algorithms, Network Decomposition, Excluded Minor} }
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