Shortest path queries in rectilinear worlds

In this paper, a data structure is given for two and higher dimensional shortest path queries. For a set of n axis-parallel rectangles in the plane, or boxes in d-space, and a fixed target, it is possible with this structure to find a shortest rectilinear path avoiding all rectangles or boxes from any point to this target. Alternatively, it is possible to find the length of the path. The metric considered is a generalization of the L1-metric and the link metric, where the length of a path is its L1-length plus some (fixed) constant times the number of turns on the path. The data structure has size O((n log n)d-1), and a query takes O(logd-1 n) time (plus the output size if the path must be reported). As a byproduct, a relatively simple solution to the single shot problem is obtained; the shortest path between two given points can be computed in time O(nd log n) for d=3, and in time O(n2) in the plane.