To exploit the similarity information hidden in the hyperlink structure of the Web, this paper introduces algorithms scalable to graphs with billions of vertices on a distributed architecture. The similarity of multistep neighborhoods of vertices are numerically evaluated by similarity functions including SimRank, a recursive refinement of cocitation, and PSimRank, a novel variant with better theoretical characteristics. Our methods are presented in a general framework of Monte Carlo similarity search algorithms that precompute an index database of random fingerprints, and at query time, similarities are estimated from the fingerprints. We justify our approximation method by asymptotic worst-case lower bounds: we show that there is a significant gap between exact and approximate approaches, and suggest that the exact computation, in general, is infeasible for large-scale inputs. We were the first to evaluate SimRank on real Web data. On the Stanford WebBase graph of 80M pages the quality of the methods increased significantly in each refinement step until step four

Personalized PageRank expresses link-based page quality around user selected pages. The only previous personalized PageRank algorithm that can serve on-line queries for an unrestricted choice of pages on large graphs is our Monte Carlo algorithm [WAW 2004]. In this paper we achieve unrestricted personalization by combining rounding and randomized sketching techniques in the dynamic programming algorithm of Jeh and Widom [WWW 2003]. We evaluate the precision of approximation experimentally on large scale real-world data and find significant improvement over previous results. As a key theoretical contribution we show that our algorithms use an optimal amount of space by also improving earlier asymptotic worst-case lower bounds. Our lower bounds and algorithms apply to the SimRank as well; of independent interest is the reduction of the SimRank computation to personalized PageRank.

To exploit the similarity information hidden in the hyperlink structure of the web, this paper introduces algorithms scalable to graphs with billions of vertices on a distributed architecture. The similarity of multi-step neighborhoods of vertices are numerically evaluated by similarity functions including SimRank [20], a recursive refinement of cocitation; PSimRank, a novel variant with better theoretical characteristics; and the Jaccard coefficient, extended to multi-step neighborhoods. Our methods are presented in a general framework of Monte Carlo similarity search algorithms that precompute an index database of random fingerprints, and at query time, similarities are estimated from the fingerprints. The performance and quality of the methods were tested on the Stanford Webbase [19] graph of 80M pages by comparing our scores to similarities extracted from the ODP directory [26]. Our experimental results suggest that the hyperlink structure of vertices within four to five steps provide more adequate information for similarity search than single-step neighborhoods.

An efficient and general graph-theoretic model (the wavelength-graph (WG)) has been proposed which enables solution of the static routing and wavelength assignment (RWA) problems in multihop wavelength routing (WR) wavelength division multiplexing (WDM) networks simultaneously, and-as a unique feature-it optimises the optical layer jointly with the electrical one. Based on the proposed WG model the problem has been formulated as an integer linear program (ILP), solved by stochastic algorithms improved by simple heuristics. The topology of the physical layer, the type of each node (e.g., OADM, OXC or EXC), the number of available wavelengths per link and the capacity of each wavelength-channel are assumed given with the aggregated traffic demand of each node-pair. The output of the optimisation is the system of wavelength paths, light paths and semi-light paths. The objective of the optimisation is to reduce resource usage at upper (electrical) layers, subject to the constrained amount of capacity of each wavelength and a limited number of wavelengths. Although the problem to be solved is NP-hard, all methods proposed give a result in a very short time.