Clustering with Few Disks to Minimize the Sum of Radii
CoRR(2023)
摘要
Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius
problem asks to cover this point set using $k$ disks with the objective of
minimizing the sum of the radii of the disks. After a long line of research on
related problems, it was finally discovered that this problem admits a
polynomial time algorithm [GKKPV~'12]; however, the running time of this
algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical
nature. A practically and structurally interesting special case of the
$k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius
problem, a near-quadratic time algorithm with expected running time $O(n^2
\log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92].
We present the first improvement of this result, namely, a near-linear time
algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n
\log^2 \log n)$ time. We generalize this result to any constant dimension $d$,
for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time
algorithm. Additionally, we give a near-quadratic time algorithm for
$3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log
n)$ time. All of these algorithms rely on insights that uncover a surprisingly
simple structure of optimal solutions: we can specify a linear number of lines
out of which one separates one of the clusters from the remaining clusters in
an optimal solution.
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