Sparse fault-tolerant BFS trees

A fault-tolerant structure for a network is required to continue functioning following the failure of some of the network's edges or vertices. This paper considers breadth-first search (BFS) spanning trees, and addresses the problem of designing a sparse fault-tolerant BFS tree, or FT-BFS tree for short, namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T? of T still contains a BFS spanning tree for (the surviving part of) G. For a source node s, a target node t and an edge e ? G, the shortest s - t path P s,t,e that does not go through e is known as a replacement path. Thus, our FT-BFS tree contains the collection of all replacement paths P s,t,e for every t ? V(G) and every failed edge e ? E(G). Our main results are as follows. We present an algorithm that for every n-vertex graph G and source node s constructs a (single edge failure) FT-BFS tree rooted at s with O(n min{Depth(s), ?n}) edges, where Depth(s) is the depth of the BFS tree rooted at s. This result is complemented by a matching lower bound, showing that there exist n-vertex graphs with a source node s for which any edge (or vertex) FT-BFS tree rooted at s has ?(n3/2) edges. We then consider fault-tolerant multi-source BFS trees, or FT-MBFS trees for short, aiming to provide (following a failure) a BFS tree rooted at each source s ? S for some subset of sources S ? V. Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every n-vertex graph and source set S ? V of size ? constructs a (single failure) FT-MBFS tree T*(S) from each source si ? S, with O(??n3/2) edges, and on the other hand there exist n-vertex graphs with source sets S ? V of cardinality ?, on which any FT-MBFS tree from S has ?(??n3/2) edges. Finally, we propose an O(logn) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there ex