Abstract
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) = ?(n1-?), then b(n) = ?T(n1+?) is necessary and sufficient. More specifically, as shown in [14], for e = 1, FT-BFS structures (with no reinforced edges) require ?(n3/2) 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(n1-?) and b(n) = O(min{1/? n1+? logn,n3/2}). 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 ?(min{n1+?,n3/2}) backup edges when r(n) = ?(n1-?) edges are reinforced. Copyright 2015 ACM.
Original language | English |
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Pages | 264-273 |
DOIs | |
State | Published - 2015 |
Event | 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2015 - Duration: 1 Jun 2015 → 1 Jun 2015 |
Conference
Conference | 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2015 |
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Period | 1/06/15 → 1/06/15 |