Rectilinear short path queries among rectangular obstacles

Abstract

Given a set of n disjoint rectangular obstacles in the plane whose edges are either vertical or horizontal, we consider the problem of processing rectilinear approximate shortest path queries between pairs of arbitrary query points. Our goal is to answer each approximate shortest path query quickly by constructing a data structure that captures path information in the obstacle-scattered plane. We present a data structure for rectilinear approximate shortest path queries that requires O( n log 2 n) time to construct and O( n log n) space. This data structure enables us to report the length of an approximate shortest path between two arbitrary query points in O( log n) time and the actual path in O( log n + L) time, where L is the number of edges of the output path. If the query points are both obstacle vertices then the length and an actual path can be reported in O(1) and O( L) time respectively. The approximation factor for the approximate shortest paths that we compute is 3. The previously best known solution to this problem requires O( n log 3 n) time and O( n log 2 n) space to build a data structure, which supports length and actual path queries respectively in O( log 2 n) and O( log 2 n + L) time (regardless of the types of query points); the approximation factor for paths between arbitrary query points is 7.