Abstract
In this paper, we investigate the following problem: “given a set \(\mathcal {S}\) of n homothetic polygons, preprocess \(\mathcal {S}\) to efficiently report all the polygons of \(\mathcal {S}\) containing a query point.” A set of polygons is said to be homothetic if each polygon in the set can be obtained from any other polygon of the set using scaling and translating operations. The problem is the counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (Chazelle, B., and Edelsbrunner, H., Linear space data structures for two types of range search. Discrete & Computational Geometry 2, 2 (1987), 113–126). We show that after preprocessing a set of homothetic polygons with constant number of vertices, the queries can be answered in \(O(\log n + k)\) optimal time, where k is the output size. The preprocessing takes \(O(n\log n)\) space and time. We also study the problem in dynamic setting where insertion and deletion operations are also allowed.
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We wish to thank anonymous referees for careful reading of the manuscript, their comments and suggestions. We believe the suggestions have helped in improving the manuscript.
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Akram, W., Saxena, S. (2023). Point Enclosure Problem for Homothetic Polygons. In: Hsieh, SY., Hung, LJ., Lee, CW. (eds) Combinatorial Algorithms. IWOCA 2023. Lecture Notes in Computer Science, vol 13889. Springer, Cham. https://doi.org/10.1007/978-3-031-34347-6_2
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