A Randomized Parallel Search Strategy

The problem of computing shortest paths in graphs is of particular interest in combinatorial optimization. Unfortunately, known parallel shortest path algorithms perform significantly more work than the best known sequential algorithms. This gap holds for both Single-Source and All-Pairs versions of this problem. We present here two parallel algorithms for the Single-Source Shortest Path problem having an efficient expected performance for two important cases of Random Graphs g(n, P(edge) = p): - a) p is a fixed constant. We give a parallel-randomized algorithm using n processors and time with “high probability” bounded by log n. - b) \(p = \frac{{c\log n}}{n}\) (for any constant c > 2). We give a parallel algorithm using n processors and time with “high probability” bounded by log2 n. This algorithm remains efficient even for \(p \in \Theta \left( {\frac{{{{\log }^k}}}{n}} \right)\) for any constant k > 1. These algorithms can be easily adapted in order to solve the All-Pairs version by using n 2 processors and the same expected time.