Searching for the closest-pair in a query translate
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In this paper, 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 plane points 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 cases: 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 O(n) space and O( n) query time, which is asymptotically optimal. When is a general convex body with a smooth boundary, we give a near-optimal data structure using O(n n) space and O(^2 n) query time. Our results settle some open questions posed by Xue et al. [SoCG 2018].
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