Bipartite Perfect Matching in Pseudo-Deterministic NC
Name
LIPIcs-ICALP-2017-87.pdf
Description
Published version
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505.45 KB
Format
Adobe PDF
Checksum (MD5)
bc9b997b60b763db75049d8336efc426
Author(s)
Goldwasser, Shafi
Grossman, Ofer
Date Issued
2017
Citation
Goldwasser, Shafi and Grossman, Ofer. 2017. "Bipartite Perfect Matching in Pseudo-Deterministic NC."
Version
Final published version
Abstract
© Shafi Goldwasser and Ofer Grossman;. 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 pseudodeterministic 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(log2n) bits whereas the previous best randomized algorithm used O(log7n), 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.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.ICALP.2017.87
