Efficient Computation of Rectilinear Geodesic Voronoi Neighbor in the Presence of Obstacles |
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Authors: | Pinaki Mitra Subhas C Nandy |
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Institution: | aDepartment of Computer Science and Engineering, Jadavpur University, Calcutta, 700 032, India;bIndian Statistical Institute, Calcutta, 700 035, India |
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Abstract: | In this paper we present an algorithm to compute the rectilinear geodesic voronoi neighbor of an arbitrary query pointqamong a setSofmpoints in the presence of a set ofnvertical 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 and is denoted by δ(α, β). The rectilinear geodesic voronoi neighbor of an arbitrary query pointq,RGVN(q) is the pointpi ∈ Ssuch that δ(q, pi) is minimum. The algorithm suggests a preprocessing of the elements of the setsSand inO((m + n)log(m + n)) time such that for an arbitrary query pointq, theRGVNquery can be answered inO(log(m + n)) time. The space required for storing the preprocessed information isO(n + m log m). If the points inSare placed on the boundary of the rectangular floor, a different technique is adopted to decrease the space complexity toO(m + n). This technique works even if the obstacles are rectangles instead of line segments. Finally, the parallelization of the preprocessing steps for the latter algorithm is suggested, which takesO(log3(m + n)) time, usingO((m + n)1.5/log2(m + n)) processors andO(log(m + n)) query time. |
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