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.ICALP.2017.87
URN: urn:nbn:de:0030-drops-74824
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7482/
Goldwasser, Shafi ;
Grossman, Ofer
Bipartite Perfect Matching in Pseudo-Deterministic NC
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 |
|
Collection: |
|
44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) |
|
Issue Date: |
|
2017 |
|
Date of publication: |
|
07.07.2017 |
