Graph Traversal Techniques and the Maximum Flow Problem in Distributed Computation

Abstract

This paper shows that graph traversal techniques have fundamental differences between serial and distributed computations in their behaviors, computational complexities, and effects on the design of graph algorithms. It has three major parts. Section I describes the computational environment for the design and description of distributed graph algorithms in terms of an architectural model for message exchanges. The computational complexity is measured in terms of the number of messages transmitted. Section II presents several distributed algorithms for the pure traversal, depth-first search, and breadth-first search techniques. Their complexities are also given. Through these descriptions are brought out some of the intrinsic differences in the behaviors and complexities of the fundamental traversal techniques between a serial and a distributed computation environment. Section III gives the distributed version of the Ford and Fulkerson algorithm for the maximum flow problem by means of depth-first search, the largest-augmentation search and breadth-first search. The complexities of these methods are found to be 0(f*|A|), 0((l + logM/(M-1)f*|V||A|) and O(|V|6), respectively, where f* is the maximum flow value of the problem, M is the maximum number of ucs in a cut, |V| is the number of vertices, and |A| is the number of arcs. Lastly, it is shown that the largest augmentation search may be a better method than the other two. This is contrary to the known results in serial computation.