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DOI: 10.4230/LIPIcs.DISC.2018.31
URN: urn:nbn:de:0030-drops-98207
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9820/
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Ghaffari, Mohsen ; Li, Jason

New Distributed Algorithms in Almost Mixing Time via Transformations from Parallel Algorithms

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Abstract

We show that many classical optimization problems - such as (1 +/- epsilon)-approximate maximum flow, shortest path, and transshipment - can be computed in tau_{mix}(G)* n^o(1) rounds of distributed message passing, where tau_{mix}(G) is the mixing time of the network graph G. This extends the result of Ghaffari et al. [PODC'17], whose main result is a distributed MST algorithm in tau_{mix}(G)* 2^O(sqrt{log n log log n}) rounds in the CONGEST model, to a much wider class of optimization problems. For many practical networks of interest, e.g., peer-to-peer or overlay network structures, the mixing time tau_{mix}(G) is small, e.g., polylogarithmic. On these networks, our algorithms bypass the Omega(sqrt n+D) lower bound of Das Sarma et al. [STOC'11], which applies for worst-case graphs and applies to all of the above optimization problems. For all of the problems except MST, this is the first distributed algorithm which takes o(sqrt n) rounds on a (nontrivial) restricted class of network graphs. Towards deriving these improved distributed algorithms, our main contribution is a general transformation that simulates any work-efficient PRAM algorithm running in T parallel rounds via a distributed algorithm running in T * tau_{mix}(G)* 2^O(sqrt{log n}) rounds. Work- and time-efficient parallel algorithms for all of the aforementioned problems follow by combining the work of Sherman [FOCS'13, SODA'17] and Peng and Spielman [STOC'14]. Thus, simulating these parallel algorithms using our transformation framework produces the desired distributed algorithms. The core technical component of our transformation is the algorithmic problem of solving multi-commodity routing - that is, roughly, routing n packets each from a given source to a given destination - in random graphs. For this problem, we obtain a new algorithm running in 2^O(sqrt{log n}) rounds, improving on the 2^O(sqrt{log n log log n}) round algorithm of Ghaffari, Kuhn, and Su [PODC'17]. As a consequence, for the MST problem in particular, we obtain an improved distributed algorithm running in tau_{mix}(G)* 2^O(sqrt{log n}) rounds.

BibTeX - Entry

@InProceedings{ghaffari_et_al:LIPIcs:2018:9820,
  author =	{Mohsen Ghaffari and Jason Li},
  title =	{{New Distributed Algorithms in Almost Mixing Time via Transformations from Parallel Algorithms}},
  booktitle =	{32nd International Symposium on Distributed Computing  (DISC 2018)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-092-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{121},
  editor =	{Ulrich Schmid and Josef Widder},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9820},
  URN =		{urn:nbn:de:0030-drops-98207},
  doi =		{10.4230/LIPIcs.DISC.2018.31},
  annote =	{Keywords: Distributed Graph Algorithms, Mixing Time, Random Graphs, Multi-Commodity Routing}
}

Keywords: Distributed Graph Algorithms, Mixing Time, Random Graphs, Multi-Commodity Routing
Seminar: 32nd International Symposium on Distributed Computing (DISC 2018)
Issue Date: 2018
Date of publication: 28.09.2018


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