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.

An information retrieval (IR) engine can rank documents based on textual proximityofkeywords within each document. In this paper we apply this notion to search across an entire database for objects that are \near " other relevant objects. Proximity search enables simple \focusing " queries based on general relationships among objects, helpful for interactive query sessions. We view the database as a graph, with data in vertices (objects) and relationships indicated by edges. Proximity is dened based on shortest paths between objects. We have implemented a prototype search engine that uses this model to enable keyword searches over databases, and we have found it very e ective for quickly nding relevant information. Computing the distance between objects in a graph stored on disk can be very expensive. Hence, we show how to build compact indexes that allow us to quickly nd the distance between objects at search time. Experiments show that our algorithms are e-cient and scale well. 1

In many fields of application shortest path finding problems in very large graphs arise. Scenarios where large numbers ofonW##O queries for shortest paths have to be processedin real-time appear for examplein tra#cinc5###HF5 systems.In such systems, the techn5Ww# con sidered to speed up the shortest pathcomputation are usually basedon precomputed incomputed5 On approach proposedoften in thiscon text is a spacereduction where precomputed shortest paths are replaced by sin## edges with weight equal to thelenOq of the corresponres shortest path.In this paper, we give a first systematic experimen tal study of such a spacereduction approach. Wein troduce theconOkW of multi-level graph decomposition Foron specificapplication scenica from the field of timetable information in public tranc ort, we perform a detailed anai ysisan experimen tal evaluation of shortest path computation based on multi-level graph decomposition.

Abstract not found

Abstract. Given a metric (V,d), a spanner is a sparse graph whose shortest-path metric approximates the distance d to within a small multiplicative distortion. In this paper, we study the problem of spanners with slack: e.g., can we find sparse spanners where we are allowed to incur an arbitrarily large distortion on a small constant fraction of the distances, but are then required to incur only a constant (independent of n) distortion on the remaining distances? We answer this question in the affirmative, thus complementing similar recent results on embeddings with slack into ℓp spaces. For instance, we show that if we ignore an ɛ fraction of the distances, we can get spanners with O(n) edgesand O(log 1) distortion for the remaining distances. ɛ We also show how to obtain sparse and low-weight spanners with slack from existing constructions of conventional spanners, and these techniques allow us to also obtain the best known results for distance oracles and distance labelings with slack. This paper complements similar results obtained in recent research on slack embeddings into normed metric spaces. 1

We perform an extensive experimental study of several dynamic algorithms for transitive closure. In particular, we implemented algorithms given by Italiano, Yellin, Cicerone et al., and two recent randomized algorithms by Henzinger and King. We propose a ne-tuned version of Italiano's algorithms as well as a new variant of them, both of which were always faster than any of the other implementations of the dynamic algorithms. We also considered simpleminded algorithms that were easy to implement and likely to be fast in practice. We tested and compared the above implementations on random inputs, on non-random inputs that are worst-case inputs for the dynamic algorithms, and on an input motivated by a real-world graph.

Abstract not found

We present a new category of graphs, recursive clique-trees K ¡ q ¢ t £ (with q ¤ 2 and t ¤ 0), which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. This family of graphs is a generalization of recent constructions with fixed degree distributions. We first compute the relevant characteristics of those graphs: diameter, clustering and power law exponent. Then we propose a labeling of the vertices of K ¡ q ¢ t £ , for any q ¤ 2 and t ¤ 0, that allows to determine a shortest path routing between any two vertices of K ¡ q ¢ t £ , only based on the above mentioned labels.

Efficient algorithms for solving the center problems in weighted cactus networks are presented. In particular, we have proposed the following algorithms for the weighted cactus networks of size n: an O(n log n) time algorithm to solve the 1center problem, an O(n log 3 n) time algorithm to solve the weighted continuous 2-center problem. We have also provided improved solutions to the general p-center problems in cactus networks. The developed ideas are then applied to solve the obnoxious 1-center problem in weighted cactus networks. 1

Input: given a weighted, connected graph, each vertex as a client having some demand. Output: the position to place one facility within the graph, which can then serve all the other clients. Objective: at which vertex we place the facility, in order to minimize the cost: • 1-median problem (Min-Sum) • 1-center problem (Min-Max)