Computing minimum-area rectilinear convex hull and L-shape |
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Authors: | Sang Won Bae Chunseok Lee Hee-Kap Ahn Sunghee Choi Kyung-Yong Chwa |
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Affiliation: | aDepartment of Computer Science and Engineering, POSTECH, Pohang, Republic of Korea;bDivision of Computer Science, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea |
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Abstract: | We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O(n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O(n2) time and O(n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space. |
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Keywords: | Rectilinear convex hull L-shape Enclosing shapes Extremal points Staircases |
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