Fault Tolerant Approximate 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 addresses the problem of designing a fault-tolerant (α, β) approximate BFS tree (or FT-ABFS tree for short), namely, a subgraph H of the given network G such that subsequent to the failure of a single edge, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v,H \ {ej}) ≤ α · dist(s, v,G \ {ej}) + β for every v ∈ V . We first consider multiplicative (α, 0) FT-ABFS structures, resilient to a failure of a single edge or vertex, and present an algorithm that for every n-vertex unweighted undirected graph G and source node s constructs a (3, 0) FT-ABFS tree rooted at s with at most 4n edges (improving by an O(log n) factor on the near-tight result of [3]). For the case of at most f -edge faults, for integer f > 1, we prove that there exists a (poly-time constructible) (3(f+1), (f+1) log n) FT-ABFS tree with O(fn) edges, resilient to the failure of up to f edges. We then consider additive (1, β) FT-ABFS structures. In contrast to the linear size of (α, 0) FT-ABFS structures, we show that for every β ∈ [1, O(log n)] there exists an n-vertex graph G with a source node s for which any (1, β) FT-ABFS tree rooted at s has Ω(n ) edges, for some function (β) ∈ (0, 1). In particular, (1, 3) FT-ABFS admit a lower bound of Ω(n) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive spanners; whereas (α, 0) FT-ABFS spanners of size O(n) exist (for α ≥ 3), their additive counterparts, (1, β) FT-ABFS spanners, are of super-linear size. Our lower bounds are comple∗Department of Computer Science and Applied Mathematics. The Weizmann Institute of Science, Rehovot, Israel. E-mail: {merav.parter,david.peleg}@ weizmann.ac.il. Supported in part by the Israel Science Foundation (grant 894/09), the United States-Israel Binational Science Foundation (grant 2008348), the I-CORE program of the Israel PBC and ISF (grant 4/11), the Israel Ministry of Science and Technology (infrastructures grant), and the Citi Foundation. †Recipient of the Google European Fellowship in distributed computing; research is supported in part by this Fellowship. mented by an upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source node s constructs a (1, 4) FT-ABFS tree rooted at s with at most O(n) edges.