The swap edges of a multiple-sources routing tree

Let T be a spanning tree of a graph G and S ⊂ V(G) be a set of sources. The routing cost of T is the total distance from all sources to all vertices. For an edge e of T, the swap edge of e is the edge f minimizing the routing cost of the tree formed by replacing e with f. Given an undirected graph G and a spanning tree T of G, we investigate the problem of finding the swap edge for every tree edge. In this paper, we propose an O(mlog∈n+n 2)-time algorithm for the case of two sources and an O(mn)-time algorithm for the case of more than two sources, where m and n are the numbers of edges and vertices of G, respectively. © 2007 Springer Science+Business Media, LLC.