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.