How to swap a failing edge of a single source shortest paths tree

Abstract

In this paper we introduce the notion of best swap for a failing edge of a single source shortest paths tree (SPT) S(r) rooted in r in a weighted graph G = (V;E). Given an edge e ∈ S(r), an edge e’ ∈\{e} is a swap edge if the swap tree Se/e’ (r) obtained by swapping e with e’ in S(r) is a spanning tree of G. A best swap edge for a given edge e is a swap edge minimizing some distance functional between r and the set of nodesdisconnected from the root after the edge e is removed. A swap algorithmwith respect to some distance functional computes a best swap edge forevery edge in S(r). We show that there exist fast swap algorithms (much faster than recomputing from scratch a new SPT) which also preserve the functionality of the aected SPT.