# Successive approximation in parallel graph algorithms

## Abstract

The notion of successive approximation is introduced in the context of parallel graph algorithms. The implementation of graph algorithms on Leighton's mesh of trees network model is considered. The implementations that have appeared so far in the literature are relatively straightforward. A common characteristic of these algorithms is that, in each iteration, for each vertex v, at most one edge is selected from the edges incident on v. This selection is based purely on local information such as the weights of the edges incident v or the labels of the neighboring vertices of v etc. As this sort of information appears on the same row of a mesh, these algorithms lend themselves to a direct implementation. In this paper we present an implementation of the open ear decomposition algorithm of Maon, Schieber and Vishkin. Some applications of open ear decomposition include parallel planarity testing, triconnectivity and 4-connectivity testing. This algorithm is different from the other algorithms considered for implementation on a mesh of trees in that a direct implementation is ruled out due to the communication problems posed by the network. Our implementation uses a technique of successive approximation. The process starts by finding an open ear decomposition of a subgraph of at most 2n edges: the edges of two edge-disjoint forests of G. Each subsequent iteration uses the decomposition from the previous step to obtain an open ear decomposition of an enlarged subgraph. This enlarged subgraph consists of the edges that received an ear label in the previous step together with at least as many new ones. Therefore the process converges in O(*log* n) iterations. The decomposition algorithm for each iteraction can be distributed on the network. The whole algorithm takes O(*log*^{3} n) time using O(n/*log* n × n/*log* n) processors. Assuming adjacency matrix representation of the graph, the achieved speedup is O(*log* n) factor off the optimal, which is the best known.

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