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Fault Tolerant Approximate BFS Structures

This paper addresses the problem of designing a {\em fault-tolerant} $(α, β)$ approximate BFS structure (or {\em 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 \emph{approximate} BFS spanning tree for (the surviving part of) $G$, satisfying $dist(s,v,H\setminus F) \leq α\cdot dist(s,v,G\setminus F)+β$ for every $v \in V$. We first consider {\em multiplicative} $(α,0)$ FT-ABFS structures resilient to a failure of a single edge 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 $4n$ edges (improving by an $O(\log n)$ factor on the near-tight result of \cite{BS10} for the special case of edge failures). Assuming at most $f$ edge failures, for constant integer $f>1$, we prove that there exists a (poly-time constructible) $(3(f+1), (f+1) \log n)$ FT-ABFS structure with $O(f n)$ edges. We then consider {\em additive} $(1,β)$ FT-ABFS structures. In contrast to the linear size of $(α,0)$ FT-ABFS structures, we show that for every $β\in [1, O(\log n)]$ there exists an $n$-vertex graph $G$ with a source $s$ for which any $(1,β)$ FT-ABFS structure rooted at $s$ has $Ω(n^{1+ε(β)})$ edges, for some function $ε(β) \in (0,1)$. In particular, $(1,3)$ FT-ABFS structures admit a lower bound of $Ω(n^{5/4})$ edges. 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 $O(n^{4/3})$ edges.