A Clustering-Based Approach to Kinetic Closest Pair

Authors Timothy M. Chan, Zahed Rahmati

Thumbnail PDF


  • Filesize: 0.54 MB
  • 13 pages

Document Identifiers

Author Details

Timothy M. Chan
Zahed Rahmati

Cite AsGet BibTex

Timothy M. Chan and Zahed Rahmati. A Clustering-Based Approach to Kinetic Closest Pair. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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 Delta over time, our KDS uses O(n log Delta) space and processes O(n^2 beta log Delta log n + n^2 beta log Delta log log Delta)) events, each in worst-case time O(log^2 n + log^2 log Delta). Here, beta is an extremely slow-growing function. The locality of the KDS is O(log n + log log Delta). Our closest pair KDS supports insertions and deletions of points. An insertion or deletion takes worst-case time O(log Delta log^2 n + log Delta log^2 log Delta). Also, we use a similar approach to provide a KDS for the all epsilon-nearest neighbors in R^d. The complexities of the previous KDSs, for both closest pair and all epsilon-nearest neighbors, have polylogarithmic factor, where the number of logs depends on dimension d. Assuming Delta 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.
  • Kinetic Data Structure
  • Clustering
  • Closest Pair
  • All Nearest Neighbors


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Pankaj K. Agarwal, Haim Kaplan, and Micha Sharir. Kinetic and dynamic data structures for closest pair and all nearest neighbors. ACM Transactions on Algorithms, 5:4:1-37, 2008. Google Scholar
  2. Julien Basch, Leonidas J. Guibas, and John Hershberger. Data structures for mobile data. Journal of Algorithms, 31:1-19, 1999. Google Scholar
  3. Julien Basch, Leonidas J. Guibas, and Li Zhang. Proximity problems on moving points. In Proceedings of the 13th Annual Symposium on Computational Geometry (SoCG'97), pages 344-351, New York, NY, USA, 1997. ACM. Google Scholar
  4. H. Brönnimann and M.T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete &Computational Geometry, 14(1):463-479, 1995. Google Scholar
  5. Timothy M. Chan and Zahed Rahmati. Approximating the minimum closest pair distance and nearest neighbor distances of linearly moving points. Computational Geometry, 2016. Google Scholar
  6. Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing (STOC'88), pages 434-444, New York, NY, USA, 1988. ACM. Google Scholar
  7. Robert J. Fowler, Michael S. Paterson, and Steven L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12(3):133-137, 1981. Google Scholar
  8. Jie Gao, Leonidas Guibas, John Hershberger, Li Zhang, and An Zhu. Discrete mobile centers. Discrete &Computational Geometry, 30(1):45-63, 2003. Google Scholar
  9. Teofilo F. Gonzalez. Covering a set of points in multidimensional space. Information Processing Letters, 40(4):181-188, 1991. Google Scholar
  10. John Hershberger. Smooth kinetic maintenance of clusters. Computational Geometry, 31(1–2):3-30, 2005. Google Scholar
  11. Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985. Google Scholar
  12. Zahed Rahmati. Simple, Faster Kinetic Data Structures. PhD thesis, University of Victoria, 2014. Google Scholar
  13. Zahed Rahmati, Mohammad Ali Abam, Valerie King, and Sue Whitesides. Kinetic k-semi-Yao graph and its applications. Computational Geometry, 2016. Google Scholar
  14. Zahed Rahmati, Mohammad Ali Abam, Valerie King, Sue Whitesides, and Alireza Zarei. A simple, faster method for kinetic proximity problems. Computational Geometry, 48(4):342-359, 2015. Google Scholar