Efficient computation of rectilinear geodesic voronoi neighbor in presence of obstacles

Abstract

In this paper, we present an algorithm to compute the rectilinear geodesic voronoi neighbor of an arbitrary query point q among a set S of m points in the presence of a set O of n vertical line segment obstacles inside a rectangular floor. The distance between a pair of points α and Β is the shortest rectilinear distance avoiding the obstacles in O and is denoted by δ(α,Β). The rectilinear geodesic voronoi neighbor of an arbitrary query point q, (RGVN(q)) is the point p _i ∃ S such that δ(q,p _i ) is minimum. The algorithm suggests a preprocessing of the elements of the set S and O in O((m+n)log(m+n)) time such that for any arbitrary query point q, the RGVN query can be answered in O(max(logm, logn)) time. The space required for storing the preprocessed information is O(n+mlogm). If the points in S are placed on the boundary of the rectangular floor, a different technique is adopted to decrease the space complexity to O(m+n). The latter algorithm works even when the obstacles are rectangles instead of line segments.