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Approximate distance oracles with improved query time
- MATHEMATICS AND COMPUTER SCIENCE DEPARTMENT, OPEN UNIVERSITY OF ISRAEL
, 2011
"... Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k ..."
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Given an undirected graph G with m edges, n vertices, and non-negative edge weights, and given an integer k ≥ 1, we show that for some universal constant c, a (2k − 1)approximate distance oracle for G of size O(kn 1+1/k) can be constructed in O ( √ km+kn 1+c/ √ k) time and can answer queries in O(k) time. We also give an oracle which is faster for smaller k. Our results break the quadratic preprocessing time bound of Baswana and Kavitha for all k ≥ 6 and improve the O(kmn 1/k) time bound of Thorup and Zwick except for very sparse graphs and small k. When m = Ω(n 1+c/ √ k) and k = O(1), our oracle is optimal w.r.t. both stretch, size, preprocessing time, and query time, assuming a widely believed girth conjecture by Erdős.
Approximate shortest paths avoiding a failed vertex : optimal data structures for unweighted graphs
"... Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data struct ..."
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Abstract. Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t> 1. report t-approximate shortest path between u and v when vertex z fails We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size nearly match the size of their best static counterparts. 1.
On Pairwise Spanners
, 2013
"... Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparses ..."
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Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f(·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f(d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u, v) in a given set P ⊆ V × V. Such P-spanners were studied before [Coppersmith,Elkin’05] only in the special case that f(·) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f(·)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P | log n) 1/4) with f(d) = d+4 log n. Alternatively, for any ε> 0, there exists a P-spanner of size O(n|P | 1/4 log n ε) with f(d) = (1 + ε)d + 4. We also consider the relevant special case that there is a critical set of nodes S ⊆ V, and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)-spanner of size O(n √ |S|) with f(d) = d + 2, and an (S × V)-spanner of size O(n √ |S | log n) with f(d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time.
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 All-Pairs Shortest Paths (APSP) problem. In the absence of fast algorithms, attempts were made to seek fast algorithms that appro ..."
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Cited by 4 (1 self)
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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 All-Pairs Shortest Paths (APSP) problem. In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diam-eter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA’96 and SICOMP’99] designed an algo-rithm that computes in Õ n2 +m
Small stretch pairwise spanners and D-spanners
, 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) P-spanner, 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) P-spanner with Õ(n|P|1/3) edges when P ⊆ V × V is arbitrary,- a (1, 2) P-spanner with Õ(n|P|1/4) 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
Editors: Natacha Portier and Thomas Wilke
"... Abstract Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f (·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f (d) in H. Spanners are very well studied in the literature. The typical goal is to construct ..."
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Abstract Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f (·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f (d) in H. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the u-v distance only for pairs (u, v) in a given set P ⊆ V × V . Such P-spanners were studied before [Coppersmith,Elkin'05] only in the special case that f (·) is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same P) and of the best known spanners (with the same f (·)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P| log n) 1/4 ) with Alternatively, for any ε > 0, there exists a P-spanner of size O(n|P| We also consider the relevant special case that there is a critical set of nodes S ⊆ V , and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)-spanner of size O(n |S|) with f (d) = d + 2, and an (S × V )-spanner of size O(n |S| log n) with f (d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time. ACM Subject Classification G.2.2 Graphs Algorithms Keywords and phrases Introduction Let G = (V, E) be an undirected unweighted graph. A subgraph H of G is a spanner with stretch function f (·) if, given any two nodes s, t ∈ V at distance δ G (s, t) in G, the distance δ H (s, t) between the same two nodes in H is at most f (δ G (α and β are the multiplicative stretch and additive stretch of the spanner, respectively). If β = 0 the spanner is called multiplicative, and if α = 1 the spanner is called purely-additive. Spanners are very well studied in the literature (see Section 1.2). The typical goal is to achieve the sparsest possible spanner for a given stretch function f (·)
Downloaded from knowledgecenter.siam.org Faster Deterministic Fully-Dynamic Graph Connectivity
"... We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in O(log2 n / log log n) amortized time and connectivity queries in O(log n / log log n) worst-case time, where n is the number of vertices of the graph. This improv ..."
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We give new deterministic bounds for fully-dynamic graph connectivity. Our data structure supports updates (edge insertions/deletions) in O(log2 n / log log n) amortized time and connectivity queries in O(log n / log log n) worst-case time, where n is the number of vertices of the graph. This improves the deterministic data structures of Holm, de Lichtenberg, and Thorup (STOC 1998, J.ACM 2001) and Thorup (STOC 2000) which both have O(log2 n) amortized update time and O(log n / log log n) worst-case query time. Our model of computation is the same as that of Thorup, i.e., a pointer machine with standard AC0 instructions. 1
