Bipartite Perfect Matching in Pseudo-Deterministic NC
Bipartite Perfect Matching in Pseudo-Deterministic NC
复制标题
伪确定性数控中的二分完美匹配
DOI：
10.4230/lipics.icalp.2017.87
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
O. Grossman
中科院分区：
文献类型：
--
作者：
S. Goldwasser;O. Grossman
提出了一种在二部图中寻找完美匹配的伪确定性NC算法。具体来说，我们的算法是一种随机并行算法，它使用多(n)处理器，多（log n）深度，多（log n）随机比特，并为每个二部输入图输出一个高概率的唯一完美匹配。也就是说，在同一个图上，对于几乎所有的随机性选择，它都返回相同的匹配。作为一个直接的结果，我们也找到了一个伪确定性NC算法构造深度优先搜索（DFS）树。我们介绍了一种计算RNC中加权图的所有最小权值完美匹配的并集的方法和一组新的权值分配，它们结合在一起可以隔离图中的唯一匹配。
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.