License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.OPODIS.2015.6
URN: urn:nbn:de:0030-drops-65971
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6597/
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Holzer, Stephan ; Pinsker, Nathan

Approximation of Distances and Shortest Paths in the Broadcast Congest Clique

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LIPIcs-OPODIS-2015-6.pdf (0.7 MB)


Abstract

We study the broadcast version of the CONGEST-CLIQUE model of distributed computing. This model operates in synchronized rounds; in each round, any node in a network of size n can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized (2+o(1))-approximation algorithm to compute all pairs shortest paths (APSP) in time ~{O}(sqrt{n}) on weighted graphs. We complement this result by proving that any randomized (2-o(1))-approximation of APSP and (2-o(1))-approximation of the diameter of a graph takes ~Omega(n) time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a (2-o(1))-approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST-CLIQUE model, as any graph-problem where each node receives a linear amount of input can be solved trivially in linear time in this model. This contrasts a recent (1+o(1))-approximation for APSP that runs in time O(n^{0.15715}) and an exact algorithm for APSP that runs in time ~O(n^{1/3}) in the unicast version of the CONGEST-CLIQUE model, a more powerful variant of the broadcast version.

This lower bound in the broadcast CONGEST-CLIQUE model is derived by first establishing a new lower bound for (2-o(1))-approximating the diameter in weighted graphs in the CONGEST model, which is of independent interest. This lower bound is then transferred to the CONGEST-CLIQUE model.

On the positive side we provide a deterministic version of Nanongkai's (2+o(1))-approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model.

BibTeX - Entry

@InProceedings{holzer_et_al:LIPIcs:2016:6597,
  author =	{Stephan Holzer and Nathan Pinsker},
  title =	{{Approximation of Distances and Shortest Paths in the Broadcast Congest Clique}},
  booktitle =	{19th International Conference on Principles of Distributed Systems (OPODIS 2015)},
  pages =	{1--16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-98-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{46},
  editor =	{Emmanuelle Anceaume and Christian Cachin and Maria Potop-Butucaru},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6597},
  URN =		{urn:nbn:de:0030-drops-65971},
  doi =		{10.4230/LIPIcs.OPODIS.2015.6},
  annote =	{Keywords: distributed computing, distributed algorithms, approximation algorithms}
}

Keywords: distributed computing, distributed algorithms, approximation algorithms
Collection: 19th International Conference on Principles of Distributed Systems (OPODIS 2015)
Issue Date: 2016
Date of publication: 13.10.2016


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