Optimal distributed all pairs shortest paths and applications

We present an algorithm to compute All Pairs Shortest Paths (APSP) of a network in a distributed way. The model of distributed computation we consider is the message passing model: in each synchronous round, every node can transmit a different (but short) message to each of its neighbors. We provide an algorithm that computes APSP in O(n) communication rounds, where n denotes the number of nodes in the network. This implies a linear time algorithm for computing the diameter of a network. Due to a lower bound these two algorithms are optimal up to a logarithmic factor. Furthermore, we present a new lower bound for approximating the diameter D of a graph: Being allowed to answer D+1 or D can speed up the computation by at most a factor D. On the positive side, we provide an algorithm that achieves such a speedup of D and computes an (1+εepsilon) multiplicative approximation of the diameter. We extend these algorithms to compute or approximate other problems, such as girth, radius, center and peripheral vertices. At the heart of these approximation algorithms is the S-Shortest Paths problem which we solve in O(|S|+D) time.