Fast partial distance estimation and applications

We study approximate distributed solutions to the weighted all-pairs-shortest-paths (APSP) problem in the CONGEST model. We obtain the following results. A deterministic (1 + ε)-approximation to APSP with running time O(ε-2nlogn) rounds. The best previously known algorithm was randomized and slower by a (log n) factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and that are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of o(n/ log n). In the relabeling model, we obtain the following results. A randomized O(k)-approximation to APSP, for any inateger k > 1, running in O(n1/2+1/k + D) rounds, where D is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximaation ratio from O(k log k) to O(k). Also, the new algorithm uses O(logn)-bit labels, which is asymptotically optimal. A randomized O(k)-approximation to APSP, for any integer k > 1, running in time O((nD)1/2 • n1/k + D) and producing compact routing tables of size labels consist of O(k log n) bits. This improves on the apaproximation ratio of (k2) for tables of that size achieved by the best previously known algorithm, which terminates faster, in O(n1/2+1/k + D) rounds. In addition, we improve on the time complexity of the best known deterministic algorithm for distributed approximate Steiner forest.