Encoding shortest paths in spatial networks

A new data structure is presented which facilitates the search for shortest paths in spatially embedded planar networks in a worst‐case time of O ( l log r ), where l is the number of edges in the shortest path to be found and r is an upper bound on the number of so‐called cross edges (these are edges connecting, for any node v , different shortest path subtrees rooted at v 's successors). The data structure is based on the idea to identify shortest path subtrees with the regions in the plane that they cover. in the worst case, The space requirement is O ( rn ), which, in general, is O(n 2 ), but for regularly shaped networks, it is expected to be only O(n√n). A decomposition of graphs into biconnected components can be used to reduce the sizes of the trees to be encoded and to reduce the complexity of the regions for these trees. The decomposition also simplifies the algorithm for computing encoding regions, which is based on minimum link paths in polygons. Approximations for region boundaries can effectively be utilized to speed up the process of shortest path reconstruction: For a realistically constrained class of networks, i.e., networks in which the ratio of the shortest path distance between any two points to the Euclidean distance between these points is bounded by a constant, it is shown that an average searching time of O ( l ) can be achieved.