In this paper, we consider a range-search variant of the closest-pair problem. Let \varGamma be a fixed shape in the plane. We are interested in storing a given set of plane points in some data structure such that for any specified translate of \varGamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important cases : when \varGamma is a polygon (possibly with holes) and when \varGamma is a general convex body whose boundary is smooth. When \varGamma is a polygon, we present a data structure using O(n) space and O(\log n) query time, which is asymptotically optimal. When \varGamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n \log n) space and O(\log^2 n) query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].
We consider a range-search variant of the closest-pair problem. Let \varGamma be a fixed shape in the plane. We are interested in storing a given set of n points in the plane in some data structure such that for any specified translate of \varGamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings : when \varGamma is a polygon (possibly with holes) and when \varGamma is a general convex body whose boundary is smooth. When \varGamma is a polygon, we present a data structure using O(n) space and O(\log n) query time, which is asymptotically optimal. When \varGamma is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n \log n) space and O(\log^2 n) query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].
R
In this paper, we
We
R
plane points in
n points in the plane in
R
cases
settings