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Vertex Fault Tolerant Additive Spanners
, 2014
"... A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a faulttolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failu ..."
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A faulttolerant structure for a network is required to continue functioning following the failure of some of the network’s edges or vertices. In this paper, we address the problem of designing a faulttolerant additive spanner, namely, a subgraph H of the network G such that subsequent to the failure of a single vertex, the surviving part of H still contains an additive spanner for (the surviving part of) G, satisfying dist(s, t,H \ {v}) ≤ dist(s, t,G \ {v}) + β for every s, t, v ∈ V. Recently, the problem of constructing faulttolerant additive spanners resilient to the failure of up to f edges has been considered [8]. The problem of handling vertex failures was left open therein. In this paper we develop new techniques for constructing additive FTspanners overcoming the failure of a single vertex in the graph. Our first result is an FTspanner with additive stretch 2 and Õ(n5/3) edges. Our second result is an FTspanner with additive stretch 6 and Õ(n3/2) edges. The construction algorithm consists of two main components: (a) constructing an FTclustering graph and (b) applying a modified pathbuying procedure suitably adopted to failure prone settings. Finally, we also describe two constructions for faulttolerant multisource additive spanners, aiming to guarantee a bounded additive stretch following a vertex failure, for every pair of vertices in S×V for a given subset of sources S ⊆ V. The additive stretch bounds of our constructions are 4 and 8 (using a different number of edges).
Improved Purely Additive FaultTolerant Spanners
"... Let G be an unweighted nnode undirected graph. A βadditive spanner of G is a spanning subgraph H of G such that distances in H are stretched at most by an additive term β w.r.t. the corresponding distances in G. A natural research goal related with spanners is that of designing sparse spanners w ..."
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Let G be an unweighted nnode undirected graph. A βadditive spanner of G is a spanning subgraph H of G such that distances in H are stretched at most by an additive term β w.r.t. the corresponding distances in G. A natural research goal related with spanners is that of designing sparse spanners with low stretch. In this paper, we focus on faulttolerant additive spanners, namely additive spanners which are able to preserve their additive stretch even when one edge fails. We are able to improve all known such spanners, in terms of either sparsity or stretch. In particular, we consider the sparsest known spanners with stretch 6, 28, and 38, and reduce the stretch to 4, 10, and 14, respectively (while keeping the same sparsity). Our results are based on two different constructions. On one hand, we show how to augment (by adding a small number of edges) a faulttolerant additive sourcewise spanner (that approximately preserves distances only from a given set of source nodes) into one such spanner that preserves all pairwise distances. On the other hand, we show how to augment some known faulttolerant additive spanners, based on clustering techniques. This way we decrease the additive stretch without any asymptotic increase in their size. We also obtain improved faulttolerant additive spanners for the case of one vertex failure, and for the case of f edge failures.