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This paper initiates the study of fault resilient network structures that mixtwo orthogonal protection mechanisms: (a) {\em backup}, namely, augmenting thestructure with many (redundant) low-cost but fault-prone components, and (b){\em reinforcement}, namely, acquiring high-cost but fault-resistantcomponents. To study the trade-off between these two mechanisms in a concretesetting, we address the problem of designing a $(b,r)$ {\em fault-tolerant} BFS(or $(b,r)$ FT-BFS for short) structure, namely, a subgraph $H$ of the network$G$ consisting of two types of edges: a set $E' \subseteq E$ of $r(n)$fault-resistant {\em reinforcement} edges, which are assumed to never fail, anda (larger) set $E(H) \setminus E'$ of $b(n)$ fault-prone {\em backup} edges,such that subsequent to the failure of a single fault-prone backup edge $e \inE \setminus E'$, the surviving part of $H$ still contains an BFS spanning treefor (the surviving part of) $G$, satisfying $dist(s,v,H\setminus \{e\}) \leqdist(s,v,G\setminus \{e\})$ for every $v \in V$ and $e \in E \setminus E'$. Weestablish the following tradeoff between $b(n)$ and $r(n)$: For every real$\epsilon \in (0,1]$, if $r(n) = {\tilde\Theta}(n^{1-\epsilon})$, then $b(n) ={\tilde\Theta}(n^{1+\epsilon})$ is necessary and sufficient.

Fault Tolerant BFS Structures: A Reinforcement-Backup Tradeoff