Parallel rectilinear shortest paths with rectangular obstacles

Abstract

Atallah, M.J. and D.Z. Chen, Parallel rectilinear shortest paths with rectangular obstacles, Computational Geometry: Theory and Applications 1 (1991) 79-113. Given a rectilinear convex polygon P having O( n) vertices and which contains n pairwise disjoint rectangular rectilinear obstacles, we compute, in parallel, a data structure that supports queries about shortest rectilinear obstacle-avoiding paths in P. That is, a query specifies a source and a destination, and the data structure enables efficient processing of the query. We construct the data structure in O(log 2 n) time, with O( n 2/log 2 n) processors in the CREW- PRAM model if all queries are such that the source and the destination are on the boundary of P, with O( n 2/log n) processors if the source is an obstacle vertex and the destination is on the boundary of P, and with O( n 2) processors if both the source and destination are arbitrary points in the plane. The data structure we compute enables one processor to obtain the path length for any pair of query vertices (of obstacles or of P) in constant time, or O(⌈ k/log n⌉) processors to retrieve the shortest path itself in logarithmic time, where k is the number of segments of that path. If the two query points are arbitrary rather than vertices, then one processor takes O(log n) time (instead of constant time) for finding the path length, while the complexity bounds for reporting an actual shortest path remain unchanged. A number of other related shortest paths problems are solved. The techniques we use involve a fast computation of staircase separators, and a scheme for partitioning the obstacles' boundaries in a way that ensures that the resulting path length matrices have a monotonicity property that is apparently absent before applying our partitioning scheme. Sequentially, the data structure can be built in O( n 2) time.