Parallel Connectivity and Minimum Spanning Trees

Abstract

Given a weighted undirected graph G with n vertices and m edges, compute a minimum spanning tree (or spanning forest) of G on a parallel random access machine (PRAM) without concurrent write capability. A minimum spanning tree of a graph is a spanning tree with the smallest possible sum of edge weights. The parallel random access machine (PRAM) is an abstract model for designing parallel algorithms and understanding the power of parallelism. In this model, processors (each being a random access machine) work in a synchronous manner and communicate through a shared memory. PARM can be further classified according to whether it is allowed for more than one processor to read and write into the same shared memory location simultaneously. The strongest model is CRCW (concurrent-read, concurrent-write) PRAM, and the weakest is EREW (exclusiveread, exclusive-write) PRAM. For an introduction to PRAM algorithms, one can refer to Karp and Ramachandran [8] and JaJa [5]. The input graph G is assumed to be given in the form of adjacency lists. Furthermore, isolated (degree-0) vertices are removed, and hence it is assumed that m n.