On parallel rectilinear obstacle- avoiding paths

We give improved space and processor complexities for the problem of computing, in parallel, a data structure that supports queries about shortest rectilinear obstacle-avoiding paths in the plane, where the obstacles are disjoint rectangles. That is, a query specifies any source and destination in the plane, and the data structure enables efficient processing of the query. We now can build the data structure with O(n²/log n) CREW PRAM processors, as opposed to the previous O(n²), and with O(n²) space, as opposed to the previous O(n²(log n)²). The time complexity remains unchanged, at O((log n)²). As before, the data structure we compute enables a query to be processed in O(log n) time, by one processor for obtaining a path length, or by O(k/log n) processors for retrieving a shortest path itself, where k is the number of segments on that path. The new ideas that made our improvement possible include a new partitioning scheme of the recursion tree, which is used to schedule the computations performed on that tree. Since a number of other related shortest paths problems are solved using this technique as a subroutine our improvement translates into a similar improvement in the complexities of these problems as well.