Point enclosure problem for homothetic polygons

In this paper, we investigate the following problem: “given a set S of n homothetic polygons, preprocess S to efficiently report all the polygons of S containing a query point.” A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling...

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Bibliographic Details
Published in:Theoretical computer science Vol. 1030; p. 115054
Main Authors: Akram, Waseem, Saxena, Sanjeev
Format: Journal Article
Language:English
Published: Elsevier B.V 13.03.2025
Subjects:
Algorithms
Data structures
Dynamic algorithms
Geometric intersection
Data structures
Algorithms
Dynamic algorithms
Geometric intersection
ISSN:0304-3975
Online Access:Get full text
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Summary:In this paper, we investigate the following problem: “given a set S of n homothetic polygons, preprocess S to efficiently report all the polygons of S containing a query point.” A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling and translation operations. The problem is a counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (1987) [9]. We show that after preprocessing a set of homothetic polygons with a constant number of vertices, queries can be answered in O(log⁡n+k) optimal time, where k is the output size. The preprocessing takes O(nlog⁡n) space and time. We also study the problem in a dynamic setting where insertion and deletion operations are allowed. The results we obtain also hold for c-oriented triangles, where c is a fixed constant.
ISSN:0304-3975
DOI:10.1016/j.tcs.2024.115054