NDLI: Kinetic and dynamic data structures for closest pair and all nearest neighbors

Content Provider	ACM Digital Library
Author	Agarwal, Pankaj K. Sharir, Micha Kaplan, Haim
Copyright Year	2008
Abstract	We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic two-dimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of $\textit{n}$ moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses $\textit{O}(\textit{n}$ log $\textit{n})$ space, and processes $O(n^{2}β_{s+2}(n)log$ $\textit{n})$ critical events, each in $O(log^{2}n)$ time. Here $\textit{s}$ is the maximum number of times where the distances between any two specific pairs of points can become equal, $β_{s}(q)$ = $λ_{s}(q)/q,$ and $λ_{s}(q)$ is the maximum length of Davenport-Schinzel sequences of order $\textit{s}$ on $\textit{q}$ symbols. The dynamic version of the problem incurs a slight degradation in performance: If $\textit{m}$ ≥ $\textit{n}$ insertions and deletions are performed, the structure still uses $\textit{O}(\textit{n}$ log $\textit{n})$ space, and processes $O(mnβ_{s}+2(n)log^{3}$ $\textit{n})$ events, each in $O(log^{3}n)$ time. Our kinetic data structure for all nearest neighbors uses $\textit{O}(\textit{n}$ $log^{2}$ $\textit{n})$ space, and processes $O(n^{2}β^{2}_{s+2}(n)log^{3}$ $\textit{n})$ critical events. The expected time to process all events is $O(n^{2}β_{s+2}^{2}(n)$ $log^{4}n),$ though processing a single event may take $Θ(\textit{n})$ expected time in the worst case. If $\textit{m}$ ≥ $\textit{n}$ insertions and deletions are performed, then the expected number of events is $O(mnβ^{2}_{s+2}(n)$ $log^{3}n)$ and processing them all takes $O(mnβ^{2}_{s+2}(n)$ $log^{4}n).$ An insertion or deletion takes $\textit{O}(\textit{n})$ expected time.
Starting Page	1
Ending Page	37
Page Count	37
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/1435375.1435379
Volume Number	5
Issue Number	1
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2008-12-12
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Kinetic data structures Closest pair Computational geometry Nearest neighbors
Content Type	Text
Resource Type	Article
Subject	Mathematics

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