Swap edges of shortest path tree in parallel

Abstract

Let G = (V, E) be a biconnected (2-edge connected), undirected graph with n vertices and m edges. A positive real weight is associated with every edge of the graph. Let d be the average depth, of a shortest path tree SG(s), rooted at s. Removal of a tree edge e = (u, v) (u is parent of v) breaks the shortest path tree into two parts, T1—the subtree containing s and T2—the sub tree rooted at v. For each tree edge e, we are required to find a non-tree edge, with one end point in T1and the other end point in T2 such that the average distance from the root s to all the nodes in the disconnected subtree T2 is minimised. The proposed parallel algorithm can be implemented on the Concurrent Read Exclusive Write (CREW) model either in (1) O((log m)3/2) time using O(m ? nd ? m(log m)1/2) operations (processor-time product), or alternatively in (2) O(log m) time using O(m + nd + m log m) operations.