Searching for the Closest-Pair in a Query Translate

Authors Jie Xue, Yuan Li, Saladi Rahul, Ravi Janardan

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Author Details

Jie Xue
  • University of Minnesota, Twin Cities, Minneapolis, MN, USA
Yuan Li
  • Facebook Inc., Seattle, WA, USA
Saladi Rahul
  • University of Illinois at Urbana-Champaign, Urbana, IL, USA
Ravi Janardan
  • University of Minnesota, Twin Cities, Minneapolis, MN, USA

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Jie Xue, Yuan Li, Saladi Rahul, and Ravi Janardan. Searching for the Closest-Pair in a Query Translate. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider a range-search variant of the closest-pair problem. Let Gamma 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 Gamma, the closest pair of points contained in the translate can be reported efficiently. We present results on this problem for two important settings: when Gamma is a polygon (possibly with holes) and when Gamma is a general convex body whose boundary is smooth. When Gamma is a polygon, we present a data structure using O(n) space and O(log n) query time, which is asymptotically optimal. When Gamma 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. at SoCG 2018.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Closest pair
  • Range search
  • Geometric data structures
  • Translation query


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