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Approximate distance oracles for unweighted graphs . . .
"... ������������ � Let be an undirected graph � on vertices, and ���������� � let denote the distance � in between two � vertices � and. Thorup and Zwick showed that for any +ve � integer, the � graph can be preprocessed to build a data-structure that can efficiently � report-approximate distance betwee ..."
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Cited by 53 (10 self)
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������������ � Let be an undirected graph � on vertices, and ���������� � let denote the distance � in between two � vertices � and. Thorup and Zwick showed that for any +ve � integer, the � graph can be preprocessed to build a data-structure that can efficiently � report-approximate distance between any pair of vertices. That is, for �������� � any, the distance � ���������� � reported satisfies The remarkable feature of this data-structure is that, ���� � for, it occupies sub-quadratic space, i.e., it does not store all-pairs distances information explicitly, and still it can answer � any-approximate distance query in constant time. They named the data-structure “oracle ” because of this feature. Furthermore the tradeoff between � stretch and the size of the data-structure is essentially optimal. In this paper we show that we can actually construct approximate distance oracles in ��������� expected time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first linear time algorithm for computing an optimal �������� � size-spanner of an unweighted graph.
A simple and linear time randomized algorithm for computing sparse . . .
"... Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Comput ..."
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Cited by 34 (5 self)
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Let G = (V, E) be an undirected weighted graph on |V | = n vertices, and |E | = m edges. A t-spanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a t-spanner of minimum size (number of edges) has been a widely studied and well motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a t-spanner of a given weighted graph. Moreover, the size of the t-spanner computed essentially matches the worst case lower bound implied by a 43 years old girth conjecture made independently by Erdős [26], Bollobás [19], and Bondy & Simonovits [21]. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information which involves growing either breadth first search trees up to θ(t)-levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory.
Faster algorithms for approximate distance oracles and all-pairs small stretch paths
- In 47th Annual IEEE Symp. on Foundations of Computer Science (FOCS
, 2006
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New Constructions of (α, β)-Spanners and Purely Additive Spanners
, 2005
"... An ¦ α § β ¨-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨-spanners of size ..."
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Cited by 27 (6 self)
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An ¦ α § β ¨-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨-spanners of size O ¦ n 1 � 1 � k ¨ and that there exist (purely additive) ¦ 1 § 2 ¨-spanners of size O ¦ n 3 � 2 ¨. However no other ¦ 1 § O ¦ 1¨� ¨-spanners are known to exist. In this paper we develop a couple new techniques for constructing ¦ α § β ¨-spanners. The first result is a purely additive ¦ 1 § 6 ¨-spanner of size O ¦ n 4 � 3 ¨. Our construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well-approximated by paths already purchased. This general approach should lead to new spanner constructions. The second result is a truly simple linear time construction of ¦ k § k � 1 ¨-spanners with size O ¦ n 1 � 1 � k ¨. In a distributed network the algorithm terminates in a constant number of rounds and has expected size O ¦ n 1 � 1 � k ¨. The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.
Distance oracles for unweighted graphs: breaking the quadratic barrier with constant additive error
, 2008
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FASTER ALGORITHMS FOR ALL-PAIRS APPROXIMATE SHORTEST PATHS IN UNDIRECTED GRAPHS
, 2006
"... Let G = (V, E) be a weighted undirected graph having non-negative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing all-pairs small stretch distances efficiently (both in terms of time ..."
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Cited by 9 (2 self)
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Let G = (V, E) be a weighted undirected graph having non-negative edge weights. An estimate ˆ δ(u, v) of the actual distance δ(u, v) between u, v ∈ V is said to be of stretch t iff δ(u, v) ≤ ˆ δ(u, v) ≤ t · δ(u, v). Computing all-pairs small stretch distances efficiently (both in terms of time and space) is a well-studied problem in graph algorithms. We present a simple, novel and generic scheme for all-pairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for all-pairs t-stretch distances for a whole range of stretch t, and also answer an open question posed by Thorup and Zwick in their
Faster approximation of distances in graphs
- IN PROC. WADS
, 2007
"... Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ∈ V reports distance no greater th ..."
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Cited by 7 (0 self)
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Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ∈ V reports distance no greater than 2dG(u, v)+h(u, v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ∈ V reports distance no greater than (1+ǫ)dG(u, v)+2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1) ǫ −3 log(nǫ −1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m √ n + n 2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for diameter approximation. Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.
Local Computation of Nearly Additive Spanners
"... An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β) ..."
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Cited by 6 (3 self)
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An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every n-node graph and integer k ≥ 1, an (α, β)-spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)-spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)-spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)-spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic
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
