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Sparse spanners vs. compact routing.
 In Proc. 23th ACM Symp. on Parallel Algorithms and Architectures (SPAA),
, 2011
"... ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ( ..."
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ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ(
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
Better approximation algorithms for the graph diameter
"... The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the AllPairs Shortest Paths (APSP) problem. In the absence of fast algorithms, attempts were made to seek fast algorithms that appro ..."
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The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the AllPairs Shortest Paths (APSP) problem. In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA’96 and SICOMP’99] designed an algorithm that computes in Õ n2 +m
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).
Small stretch pairwise spanners and Dspanners
, 2014
"... An (α, β)spanner of an undirected unweighted connected graph G = (V,E) is a subgraph H such that: dH(u, v) ≤ α · dG(u, v) + β, for all pairs (u, v) ∈ V × V, where dH(u, v) and dG(u, v) are the distances between u and v in H and G respectively. The quantities α and β are non negative real numbers ..."
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An (α, β)spanner of an undirected unweighted connected graph G = (V,E) is a subgraph H such that: dH(u, v) ≤ α · dG(u, v) + β, for all pairs (u, v) ∈ V × V, where dH(u, v) and dG(u, v) are the distances between u and v in H and G respectively. The quantities α and β are non negative real numbers and are called the multiplicative stretch and additive stretch of the spanner respectively. If α = 1, the spanner is called additive. In this report, we focus our attention to additive spanners. Additive spanners are well studied. We study a natural generalization of the additive spanner problem where we look to approximate the distances of only a specified set of pairs of nodes. Given a graph G = (V,E) and a set P ⊆ V ×V, an (α, β) Pspanner, or a pairwise spanner, of G is a subgraph H such that dH(u, v) ≤ α · dG(u, v) + β for all (u, v) ∈ P. We obtain polynomial time constructions for the following pairwise spanners: a (1, 2) Pspanner with Õ(nP1/3) edges when P ⊆ V × V is arbitrary, a (1, 2) Pspanner with Õ(nP1/4) edges when P = S × V for some S ⊆ V. In the special case when P contains exactly those pairs of nodes which are at a distance
Very Sparse Additive Spanners and Emulators
, 2015
"... We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation,” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edges ..."
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We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation,” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edges with Õ(n1/2−/2) additive distortion. 2. Every graph has an emulator on Õ(n1+) edges with Õ(n1/3−2/3) additive distortion whenever ∈ [0, 1 5 3. Every graph has a spanner on Õ(n1+) edges with Õ(n2/3−5/3) additive distortion whenever ∈ [0, 1 4 Our first spanner has the new best known asymptotic edgeerror tradeoff for additive spanners whenever ∈ [0, 1 7 Our second spanner has the new best tradeoff whenever