Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal.

We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use\Omega\Gamma n) local space at some vertex. 1 Introduction Let G = (V; E) with jV j = n be a labeled undirected network. Assuming that a positive cost, or distance is assigned with each edge, the stretch of path p(u; v) from node u to node v is defined as jp(u;v)j jd(u;v)j , where jd(u; v)j is the length of the shortest u \Gamma v path. The approximate all-pairs shortest path problem involves a tradeoff of stretch against time-- short paths with stretch bounded by a constant are com...

Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe g) time algorithm APASP 2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an g) time algorithm APASP k for computing all distances in G with an additive one-sided error of at most k.

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We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves...

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms. The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in ~ O(n 2+ ) time, where satisfies the equation !(1; ; 1) = 1 + 2 and !(1; ; 1) is the exponent of the multiplication of an n \Theta n matrix by an n \Theta n matrix. The currently best available bounds on !(1; ; 1), obtained by Coppersmith and Winograd, and by Huang and Pan, imply that ! 0:575. The running time of our algorithm is therefore O(n 2:575 ). Our algorithm improves on the ~ O(n (3+!)=2 ) time algorithm, where ! = !(1; 1; 1) ! 2:376 is the usual exponent of matrix multiplication, obtained by Alon, Galil and Margalit, whose running time is only known to be O(n 2:688 ). The second

Let G = (V; E) be a weighted undirected graph. A path between u; v 2 V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G. It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n \Theta n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH 2 , runs in ~ O(n 3=2 m 1=2 ) time and finds stretch 2 paths. The second algorithm, STRETCH 7=3 , runs in ~ O(n 7=3 ) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH 3 , runs in ~ O(n 2 ) and finds stretch 3 paths. Our algorithms are simpler, more efficient and more accurate than the previously best algorithms ...

Given a geometric t-spanner graph G in E d with n points and m edges, with edge lengths that lie within a polynomial (in n) factor of each other. Then, after O(m+n log n) preprocessing, we present an approximation scheme to answer (1+")- approximate shortest path queries in O(1) time. The data structure uses O(n log n) space.

Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For graphs exhibiting the small world phenomenon, our method estimates the centrality of all vertices with high probability within a (1 + #) factor in near-linear time. 1 Introduction In social network analysis, the vertices of a graph represent agents in a group and the edges represent relationships, such as communication or friendship. The idea of applying graph theory to analyze the connection between the structural centrality and group process was introduced by Bavelas [4]. Various measurement of centrality [7, 14, 15] have been proposed for analyzing communication activity, control, or independence within a social network. We are particularly interested in closeness centrality [5, 6, 24]...