In this paper the authors propose an approximation method to answer shortest path queries in graphs, based on hierarchical random sampling and Voronoi duals. The lowest level of the hierarchy stores the initial graph. At each higher level, we compute a simplification of the graph on the level below, by selecting a constant fraction of nodes. Edges are generated as the Voronoi dual within the lower level, using the selected nodes as Voronoi sites. This hierarchy allows for fast computation of

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... roximate shortest paths for general graphs. The time-quality trade-off decision can be made at query time. We provide bounds on the approximation ratio of the path lengths. Xuyan QIN, Yuanyuan WANG, Huapu LU (2005) One of the key issues of stochastic traffic assignment (STA) is to make the flow pattern consistent with the practical results. In this paper, the author makes a summary of STOCH algorithm and puts forward a new algorithm to solve the problem of stochastic traffic assignment called k-shortest-paths-based method. First, the paper discussed the widely used STOCH algorithm, especially for "single-pass" and "double-pass" procedures. Case studies with multi-OD pairs are analyzed to demonstrate the steps and advantages in "single-pass" method. Then, the paper presents the "k-shortest-paths-based method", which not only solves all the stochastic numerical models and improve some drawbacks existed in the current STOCH algorithms, but also posses merits of flexibility, though with a modestly higher computational requirements. The detail explanation on the design idea and steps of the algorithm is given. Finally, the time complexity of various algorithms mentioned in this paper is illustrated. Rachit Agarwal, P Brighten Godfrey & Sariel Har-Peled (2003) An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k − 1 paths must require space Ω (n1+1/k) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω (n1/k). The author present data structures that, for sparse graphs, substantially break that lower bound barrier at the expense of higher query time. For instance, general graphs require O(n3/2 ) space and constant query time for stretch 3 paths. For the realistic scenario of a graph with average degree Θ (log n), special cases of our data structure. Gilbert Laporte (2001)