Fault Tolerant Approximate BFS Structures

Abstract

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 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 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.We first consider multiplicative (α, 0) FT-ABFS structures resilient to a failure of a single edge or vertex, and present an algorithm that given an n-vertex unweighted undirected graph G and a source s constructs a (3,0) FT-ABFS structure rooted at s with at most 3n edges (improving by an O(logn) factor on the near-tight result of [3]). Assuming at most f edge failures, for constant integer f > 1, we prove that there exists a (poly-time constructible) (3(f +1), (f +1) logn) FT-ABFS structure with O(fn) 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(logn)] there exists an n-vertex graph G with a source s for which any (1, β) FT-ABFS structure rooted at s has Ω(n1+∊(β)) edges, for some function ∊(β) ∊ (0,1). In particular, (1,3) FT-ABFS structures admit a lower bound of Ω(n5/4) edges. These lower bounds demonstrate an interesting dichotomy between multiplicative and additive spanners; whereas (α, 0) FT-ABFS structures of size O(n) exist (for α ≥ 3), their additive counterparts, (1,β) FT-ABFS structures, are of super-linear size. Our lower bounds are complemented by an upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a (1,4) FT-ABFS structure rooted at s with at most O(n4/3) edges.