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DOI: 10.4230/LIPIcs.ICALP.2017.87
URN: urn:nbn:de:0030-drops-74824
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7482/
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Goldwasser, Shafi ; Grossman, Ofer

Bipartite Perfect Matching in Pseudo-Deterministic NC

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LIPIcs-ICALP-2017-87.pdf (0.5 MB)


Abstract

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph. We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art. Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm.

BibTeX - Entry

@InProceedings{goldwasser_et_al:LIPIcs:2017:7482,
  author =	{Shafi Goldwasser and Ofer Grossman},
  title =	{{Bipartite Perfect Matching in Pseudo-Deterministic NC}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{87:1--87:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7482},
  URN =		{urn:nbn:de:0030-drops-74824},
  doi =		{10.4230/LIPIcs.ICALP.2017.87},
  annote =	{Keywords: Parallel Algorithms, Pseudo-determinism, RNC, Perfect Matching}
}

Keywords: Parallel Algorithms, Pseudo-determinism, RNC, Perfect Matching
Seminar: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 06.07.2017


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