Fault-Tolerant approximate BFS structures

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 article addresses the problem of designing a fault-tolerant (α, β) approximate BFS structure (or FT-ABFS structure for short), namely, a subgraph H of the network G such that subsequent to the failure of some subset F of edges or vertices, the surviving part of H (namely, H \ F) still contains an approximate BFS spanning tree for (the surviving part of) G, satisfying dist(s, v, H \ F ) ≤ α · dist(s, v, G \ F ) + β for every v ∈ V. Our first result is an algorithm that given an n-vertex unweighted undirected graph G and a source s constructs a multiplicative (3, 0) FT-ABFS structure rooted at s resilient to a failure of a single edge with at most 4n edges (improving by an O(log n) factor on the near-tight result of Baswana and Khanna (2010) for the special case of edge failures). This was recently improved to 2n edges by Bilò et al. (2014). Next, we consider the multiple edge faults case, for a constant integer f > 1, we prove that there exists a (polynomial-time constructible) (3 f , f log n) FT-ABFS structure with O(f n) edges that is resilient against f faults. We also show the existence of a (3 f + 1, 0) FT-ABFS structure with O(f log^f n · n) edges. We then consider additive (1, β) FT-ABFS structures and demonstrate an interesting dichotomy between multiplicative and additive spanners. In contrast to the linear size of (α, 0) FT-ABFS structures, we show that for every n, there exist δ, ϵ > 0, and n-vertex graphs G with a source s for which any (1, n^δ ) FT-ABFS structure rooted at s has Ω (n⁷/6−^ϵ ) edges. For the case of additive stretch 3, we show that (1, 3) FT-ABFS structures admit a lower bound of Ω (n⁵/⁴ ) edges.