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 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 worstcase time O(log^2 n + log^2 log Delta). Here, beta is an extremely slowgrowing 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 worstcase 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 epsilonnearest neighbors in R^d.
The complexities of the previous KDSs, for both closest pair and all epsilonnearest 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.
BibTeX  Entry
@InProceedings{chan_et_al:LIPIcs:2016:6050,
author = {Timothy M. Chan and Zahed Rahmati},
title = {{A ClusteringBased Approach to Kinetic Closest Pair}},
booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)},
pages = {28:128:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770118},
ISSN = {18688969},
year = {2016},
volume = {53},
editor = {Rasmus Pagh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6050},
URN = {urn:nbn:de:0030drops60508},
doi = {10.4230/LIPIcs.SWAT.2016.28},
annote = {Keywords: Kinetic Data Structure, Clustering, Closest Pair, All Nearest Neighbors}
}
Keywords: 

Kinetic Data Structure, Clustering, Closest Pair, All Nearest Neighbors 
Collection: 

15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016) 
Issue Date: 

2016 
Date of publication: 

22.06.2016 