Point Enclosure Problem for Homothetic Polygons

TL;DR

This paper addresses the homothetic point enclosure problem for triangles with sides parallel to three fixed directions, providing an efficient data structure for fast query responses.

Contribution

It introduces an $O(n ext{log}n)$ space data structure supporting $O( ext{log}n + k)$ query time for homothetic point enclosure, applicable to triangles and polygons.

Findings

Supports efficient point enclosure queries for homothetic triangles.

Achieves $O(n ext{log}n)$ preprocessing and space complexity.

Extends results to homothetic polygons.

Abstract

In this paper, we investigate the homothetic point enclosure problem: given a set $S$ of $n$ triangles with sides parallel to three fixed directions, find a data structure for $S$ that can report all the triangles of $S$ that contain a query point efficiently. The problem is "inverse" of the homothetic range search problem. We present an $O(n\log n)$ space solution that supports the queries in $O(\log n + k)$ time, where $k$ is the output size. The preprocessing time is $O(n\log n)$. The same results also hold for homothetic polygons.