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.

Let G = (V, E) be a weighted undirected graph with |V | = n and |E | = m. An estimate ˆ δ(u, v) of the 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 distances of small stretch efficiently is a well-studied problem in graph algorithms. The most efficient algorithms known here are the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k ≥ 1, Thorup and Zwick in [16] gave an O(kmn 1/k) algorithm to construct a data structure of size O(kn 1+1/k) to answer approximate distance queries. For a query (u, v) ∈ V × V, the distance returned is of stretch at most 2k−1. The query answering time is O(k), which is essentially a constant since we are interested in small-stretch distances, or equivalently, small values of k. But for small values of k, the time to construct the oracle is rather high. The case k = 2 is particularly interesting and the Thorup-Zwick algorithm takes O(m √ n) time, which could be as large as Θ(n 5/2). Here we present an O(n 2 log n) algorithm to construct such a data structure of size O(kn 1+1/k) for all integers k ≥ 2. Our query answering time is O(k) for k> 2 and Θ(log n) for k = 2. We obtain these results with a new generic scheme for all-pairs approximate shortest paths. Using this scheme, we also design faster algorithms for all-pairs t-stretch distances for t = 2, 7/3, 3. 1.

Abstract. 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

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Abstract. 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

In this paper, we present an algorithm to compute all pairs optimized shortest paths in an unweighted and undirected graph with some additive error of at most 2.This algorithm can be extended for weighted graph also but it will not work for directed graph due to absence of commutative property. The algorithm runs in n 5/2) times, where n is the number of vertices in the graph. This algorithm is much simpler than the existing algorithms. A study of upper bounds on the size of a maximal independent set of such graphs has been performed.