Efficient algorithms for shortest distance queries on special classes of polygons

The problem of finding a rectilinear minimum bend path (RMBP) between two designated points inside a rectilinear polygon has applications in robotics and motion planning. In this paper, we present efficient algorithms to solve the query version of the RMBP problem for special classes of rectilinear polygons given their visibility graphs. Specifically, we show that given an unweighted graph G = (V, E), with ¦V¦ = N and ¦E¦ = M, algorithms to preprocess G in linear space and time such that the shortest distance queries — queries asking for the distance between any pair of nodes in the graph — can be answered in constant time and space are presented in this paper. For the case of a chordal graph G, our algorithms give a distance which is at most one away from the actual shortest distance. When G is a K-chordal graph, our algorithm produces an exact shortest distance in O(K) time. We also present a non-trivial parallel implementation of the sequential preprocessing algorithm for the CREW-PRAM model which runs in O(log² N) time using O(N + M) processors. After the preprocessing, we can answer the queries in constant time using a single processor.