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 $e' \in E\backslash \left\{ e \right\}$ is a swap edge if the swap tree $S_{e\backslash e'} \left( r \right)$ 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 nodes disconnected from the root after the edge e is removed. A swap algorithm with respect to some distance functional computes a best swap edge for every 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 affected SPT.

This research was partially supported by the CHOROCHRONOS TMR Program of the European Community. The work of the third author was partially supported by grant “Combinatorics and Geometry” of the Swiss National Science Foundation.