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Algorithmica

, Volume 80, Issue 10, pp 2742–2756 | Cite as

A Clustering-Based Approach to Kinetic Closest Pair

  • Zahed Rahmati
  • Timothy M. Chan
Article
  • 179 Downloads

Abstract

Given a set P of n moving points in fixed dimension d, where the trajectory of each point is a polynomial of degree bounded by some constant, we present a kinetic data structure (KDS) for maintenance of the closest pair on P. Assuming the closest pair distance is between 1 and \(\varDelta \) over time, our KDS uses \(O(n \log \varDelta )\) space and processes \(O(n^{2} \beta \log \varDelta \log n + n^{2} \beta \log \varDelta \log \log \varDelta )\) events, each in worst-case time \(O(\log ^2 n + \log ^2 \log \varDelta )\). Here, \(\beta \) is an extremely slow-growing function. The locality of the KDS is \(O(\log n + \log \log \varDelta )\). Our closest pair KDS supports insertions and deletions of points. An insertion or deletion takes worst-case time \(O(\log \varDelta \log ^2 n + \log \varDelta \log ^2 \log \varDelta )\). Also, we use a similar approach to provide a KDS for the all \(\varepsilon \)-nearest neighbors in \(\mathbb {R}^d\). The complexities of the previous KDSs, for both closest pair and all \(\varepsilon \)-nearest neighbors, have polylogarithmic factors, where the number of logs depends on dimension d. Assuming \(\varDelta \) is polynomial in n, our KDSs obtain improvements on the previous KDSs. Our solutions are based on a kinetic clustering on P. Though we use ideas from the previous clustering KDS by Hershberger, we simplify and improve his work.

Keywords

Kinetic data structure Clustering Closest pair All nearest neighbors Moving points 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran
  2. 2.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignChampaignUSA

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