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Approximate $k$-nearest neighbor graph on moving points | ||
| Transactions on Combinatorics | ||
| مقاله 16، دوره 12، شماره 2، شهریور 2023، صفحه 65-72 اصل مقاله (509.62 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2022.130533.1943 | ||
| نویسنده | ||
| Zahed Rahmati* | ||
| Department of Mathematics and Computer Science, Amirkabir University of Technology, P.O.Box 15875-4413, Tehran, Iran. | ||
| چکیده | ||
| In this paper, we introduce an approximation for the $k$-nearest neighbor graph ($k$-NNG) on a point set $P$ in $\mathbb{R}^d$. For any given $\varepsilon>0$, we construct a graph, that we call the \emph{approximate $k$-NNG}, where the edge with the $i$th smallest length incident to a point $p$ in this graph is within a factor of $(1+\varepsilon)$ of the length of the edge with the $i$th smallest length incident to $p$ in the $k$-NNG. For a set $P$ of $n$ moving points in $\mathbb{R}^d$, where the trajectory of each point $p\in P$ is given by $d$ polynomial functions of constant bounded degree, where each function gives one of the $d$ coordinates of $p$, we compute the number of combinatorial changes to the approximate $k$-NNG, and provide a kinetic data structure (KDS) for maintenance of the edges of the approximate $k$-NNG over time. Our KDS processes $O(kn^2\log^{d+1} n)$ events, each in time $O(\log^{d+1}n)$. | ||
| کلیدواژه‌ها | ||
| Approximation؛ $k$-Nearest Neighbor Graph؛ Moving Points | ||
| مراجع | ||
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