Fault Tolerant Approximate BFS Structures with Additive Stretch

Abstract

This paper addresses the problem of designing a $$\beta $$ -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 $$\mathrm{dist}(s,v,H{\setminus } \{e\}) \le \mathrm{dist}(s,v,G{\setminus } \{e\})+\beta $$ for every $$v \in V$$ . It was shown in Parter and Peleg (SODA, 2014), that for every $$\beta \in [1, O(\log n)]$$ there exists an n-vertex graph G with a source s for which any $$\beta $$ -additive FT-ABFS structure rooted at s has $$\Omega (n^{1+\epsilon (\beta )})$$ edges, for some function $$\epsilon (\beta ) \in (0,1)$$ . In particular, 3-additive FT-ABFS structures admit a lower bound of $$\Omega (n^{5/4})$$ 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.