Fast parallel algorithms for testing k-connectivity of directed and undirected graphs

W. Liang, B. McKay
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引用次数: 3

Abstract

It appears that no NC algorithms have previously appeared for testing a directed graph for k-edge connectivity or k-vertex connectivity, even for fixed k>1. Using an elementary flow method we give such algorithms, with time complexity O(k log n) using nP(n,m) or (n+k/sup 2/)P(n,m) processors, respectively. Here, n is the number of vertices, m is the number of edges, P(n,m) is the number of processors needed to find some path in time O(log n) time between two specified vertices in a directed graph with O(n) vertices and O(m) edges, and the computation model is a CRCW PRAM. These algorithms of course apply also to undirected graphs, but using sparse certificates we can improve the factors P(n,m) to P(n,kn) for both types of connectivity. This is better in time by a factor of O(k) over previous algorithms for undirected graphs. We also note that edge connectivity is NC-reducible to vertex connectivity even if k is not fixed.<>
有向图和无向图k-连通性的快速并行算法
似乎以前没有NC算法出现用于测试有向图的k边连通性或k顶点连通性,即使是固定k>1。使用基本流方法,我们给出了这样的算法,时间复杂度为O(k log n),分别使用nP(n,m)或(n+k/sup 2/)P(n,m)个处理器。其中,n为顶点数,m为边数,P(n,m)为具有O(n)个顶点和O(m)条边的有向图中,在O(log n)时间内找到两个指定顶点之间的路径所需的处理器数,计算模型为CRCW PRAM。这些算法当然也适用于无向图,但是使用稀疏证明我们可以将因子P(n,m)提高到P(n,kn),用于两种类型的连通性。这比以前的无向图算法在时间上要好0 (k)倍。我们还注意到,即使k不固定,边缘连通性也可以nc约化为顶点连通性。
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