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DOI: 10.4230/LIPIcs.ITCS.2025.45

Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs

Authors: Jinfeng Dou, Thorsten Götte, Henning Hillebrandt, Christian Scheideler, and Julian Werthmann

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

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.

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Jinfeng Dou, Thorsten Götte, Henning Hillebrandt, Christian Scheideler, and Julian Werthmann. Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 45:1-45:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@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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DOI: 10.4230/LIPIcs.OPODIS.2024.36

Distributed Branching Random Walks and Their Applications

Authors: Vijeth Aradhya, Seth Gilbert, and Thorsten Götte

Published in: LIPIcs, Volume 324, 28th International Conference on Principles of Distributed Systems (OPODIS 2024)

Abstract

In recent years, the explosion of big data and analytics has necessitated distributed storage and processing with several compute nodes (e.g., multiple datacenters). These nodes collaboratively perform parallel computation, where the data is typically partitioned across these nodes to ensure scalability, redundancy and load-balancing. But the nodes may not always be co-located; in many cases, they are part of a larger communication network. Since those nodes only need to communicate among themselves, a key challenge is to design efficient routes catered to that subnetwork. In this work, we initiate the study of distributed sampling and routing problems for subnetworks in any well-connected network. Given any network G = (V, E) with mixing time τ_mix, consider the canonical problem of permutation routing [Ghaffari, Kuhn and Su, PODC 2017] that aims to minimize both congestion and dilation of the routes, where the demands (i.e., set of source-terminal pairs) are such that each node sends or receives number of messages proportional to its degree. We show that the permutation routing problem, when demands are restricted to any subset S ⊆ V (i.e., subnetwork), can be solved in exp(O(√(log|S|))) ⋅ Õ(τ_mix) rounds (where Õ(⋅) hides polylogarithmic factors of |V|). This means that the running time depends subpolynomially on the subnetwork size (i.e., not on the entire network size). The ability to solve permutation routing efficiently immediately implies that a large class of parallel algorithms can be simulated efficiently on the subnetwork. As a prerequisite to constructing efficient routes, we design and analyze distributed branching random walks that distribute tokens started by the nodes in the subnetwork. At a high-level, these algorithms operate by always moving each token according to a (lazy) simple random walk, but also branching a token into multiple tokens at some specified intervals; ultimately, if a node starts a branching walk, with its id in a token, then by the end of execution, several tokens with its id would be randomly distributed among the nodes. As these random walks can be started by many nodes, a crucial challenge is to ensure low-congestion, which is a primary focus of this paper.

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Vijeth Aradhya, Seth Gilbert, and Thorsten Götte. Distributed Branching Random Walks and Their Applications. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aradhya_et_al:LIPIcs.OPODIS.2024.36,
  author =	{Aradhya, Vijeth and Gilbert, Seth and G\"{o}tte, Thorsten},
  title =	{{Distributed Branching Random Walks and Their Applications}},
  booktitle =	{28th International Conference on Principles of Distributed Systems (OPODIS 2024)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-360-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{324},
  editor =	{Bonomi, Silvia and Galletta, Letterio and Rivi\`{e}re, Etienne and Schiavoni, Valerio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2024.36},
  URN =		{urn:nbn:de:0030-drops-225723},
  doi =		{10.4230/LIPIcs.OPODIS.2024.36},
  annote =	{Keywords: Distributed Graph Algorithms, Random Walks, Permutation Routing}
}

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Documents authored by Götte, Thorsten

Distributed and Parallel Low-Diameter Decompositions for Arbitrary and Restricted Graphs

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Distributed Branching Random Walks and Their Applications

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