A Clustering-Based Approach to Kinetic Closest Pair

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.