Fault Tolerant Approximate BFS Structures with Additive Stretch

This paper addresses the problem of designing a β-additive fault-tolerant approximate BFS (or FT-ABFS for short) structure, namely, a subgraph H of the network G such that subsequent to the failure of a single edge e, the surviving part of H still contains an approximate BFS spanning tree for (the surviving part of) G, whose distances satisfy dist (s, v, H\ { e}) ≤ dist (s, v, G\ { e}) + β for every v∈ V. It was shown in Parter and Peleg (SODA, 2014), that for every β∈ [1 , O(log n)] there exists an n-vertex graph G with a source s for which any β-additive FT-ABFS structure rooted at s has Ω (n^1+ϵ(β)) edges, for some function ϵ(β) ∈ (0 , 1). In particular, 3-additive FT-ABFS structures admit a lower bound of Ω (n^{5 / 4}) edges. In this paper we present the first upper bound, showing that there exists a poly-time algorithm that for every n-vertex unweighted undirected graph G and source s constructs a 4-additive FT-ABFS structure rooted at s with at most O(n^{4 / 3}) edges. The main technical contribution of our algorithm is in adapting the path-buying strategy used in Baswana et al. (ACM Trans Algorithms 7:A5, 2010) and Cygan et al. (Proceedings of the 30th symposium on theoretical aspects of computer science, pp 209–220, 2013) to failure-prone settings.