# 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.

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