Parallel search algorithms for graphs and trees

Abstract

This paper surveys parallel algorithms for tree traversal and graph search techniques. The principal model of computation considered here is the Parallel Random Access Machine. In this model, each processor can access any cell of a shared memory in unit time. Thus, the issue of interprocessor communication do not get intermixed with the logical structure of parallel algorithms. The paper is organized into four major sections. Section 2 deals with tree traversals. There are three main algorithms that differ completely from one another in approach. In absolute terms, however, all require a specific input form or designed to preprocess the input in desired form. Section 3 presents parallel algorithms for breadth-first search of general graphs. There are two major techniques for developing efficient parallel algorithms for breadth-first search of a graph. One is based on computation of shortest paths in a uniformly positive edge weighted graph. The other is based on tree merging technique. Section 4 is devoted to parallel algorithms for depth-first search of graphs. Considering the importance of depth-first search as a general paradigm for algorithmic solution of several graph problem, there have been lot of efforts to design efficient parallel algorithms for depth-first search. But the success so far has been restricted to small classes of graphs like directed acyclic graphs (DAG) or planar graphs. For DAGs, there are two different techniques to obtain depth-first spanning trees or forests, viz., lexicographic sorting of paths and tree merging. In the case of the planar graph, the technique is to recursively divide the input graph into smaller graphs by path separators, and then run the algorithm in parallel on smaller graphs produced by the separators. Section 5 talks about car decomposition search of graphs and efficient parallel algorithms for the problem. In fact, ear decomposition search is considered as an alternative to depth-first search in designing parallel algorithms for complicated graph problems. The strategy used for car decomposition of a graph is almost identical to that used for orientation of the edges of a connected, bridgeless undirected graph so that the resultant graph is strongly connected.