Fault tolerant BFS structures: A reinforcement-backup tradeoff

This paper initiates the study of fault resilient network structures that mix two orthogonal protection mechanisms: (a) backup, namely, augmenting the structure with many (redundant) low-cost but fault-prone components, and (b) reinforcement, namely, acquiring high-cost but fault-resistant components. To study the trade-off between these two mechanisms in a concrete setting, we address the problem of designing a (b, r) 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' ⊆ E of r(n) fault-resistant reinforcement edges, which are assumed to never fail, and a (larger) set E(H)\E' of b(n) fault-prone backup edges, such that subsequent to the failure of a single fault-prone backup edge e ∈ E \ E', the surviving part of H still contains a BFS spanning tree for (the surviving part of) G, satisfying dist(s, v, H \ {e}) ≤ dist(s,v, G \ {e}) for every v ∈ V and e G E\E'. We establish the following tradeoff: For every real e G (0,1], if r(n) = Θ(n^1-∈), then b(n) = ΘT(n^1+∈) is necessary and sufficient. More specifically, as shown in [14], for e = 1, FT-BFS structures (with no reinforced edges) require Θ(n^3/2) edges, and this is sufficient. At the other extreme, if ∈ = 0, then n - 1 reinforced edges suffice (with no need for backup). Here, we present a polynomial time algorithm that given an undirected graph G = (V, E), a source vertex s and a real ∈ ∈ (0,1], constructs a (b(n), r(n)) FT-BFS with r(n) = O(n^1-∈) and b(n) = O(min{1/∈ · n^1+∈ logn,n^3/2}). We complement this result by providing a nearly matching lower bound, showing that there are n-vertex graphs for which any (b(n),r(n)) FT-BFS structure requires Ω(min{n^1+∈,n^3/2}) backup edges when r(n) = Ω(n^1-∈) edges are reinforced.