Searching for the closest-pair in a query translate
Jie Xue, Yuan Li, Saladi Rahul, Ravi Janardan

TL;DR
This paper develops efficient data structures for quickly finding the closest pair of points within any translated shape in the plane, addressing both polygonal and smooth convex bodies, and resolving open questions in the field.
Contribution
It introduces optimal and near-optimal data structures for range-search closest pair problems for polygons and smooth convex bodies, improving query efficiency.
Findings
Polygon case: $O(n)$ space, $O( ext{log } n)$ query time
Convex body case: $O(n ext{ log } n)$ space, $O( ext{log}^2 n)$ query time
Results settle open questions from prior research
Abstract
We consider a range-search variant of the closest-pair problem. Let be a fixed shape in the plane. We are interested in storing a given set of points in the plane in some data structure such that for any specified translate of , the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when is a polygon (possibly with holes) and when is a general convex body whose boundary is smooth. When is a polygon, we present a data structure using space and query time, which is asymptotically optimal. When is a general convex body with a smooth boundary, we give a near-optimal data structure using space and query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgorithms and Data Compression · Data Management and Algorithms · Computational Geometry and Mesh Generation
