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Clustering with Few Disks to Minimize the Sum of Radii

Mikkel Abrahamsen, Sarita de Berg, Lucas Meijer, André Nusser, Leonidas Theocharous

TL;DR

This work investigates the $k$-MinSumRadius clustering problem in the plane and beyond, focusing on small $k$ where practical exact algorithms are attainable. The authors reveal a surprisingly simple separator structure: in an optimal solution there exists a line from a linear set of candidates that separates one cluster from the rest, enabling efficient evaluation via dynamic minimum enclosing disks/balls. They achieve near-linear time for $2$-MinSumRadius in the plane with $O(n \log^2 n \log^2 \log n)$ expected time and extend to constant dimensions with $O\left(n^{2-1/(\lceil d/2\rceil+1)+\varepsilon}\right)$ time; they also obtain a near-quadratic plane algorithm for $3$-MinSumRadius with $O(n^2 \log^2 n \log^2 \log n)$ expected time. The core idea—limiting the search to a linear set of separators and dynamically maintaining MECs—offers a path toward more scalable exact clustering in geometric settings and highlights separator-based techniques for related problems.

Abstract

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.

Clustering with Few Disks to Minimize the Sum of Radii

TL;DR

This work investigates the -MinSumRadius clustering problem in the plane and beyond, focusing on small where practical exact algorithms are attainable. The authors reveal a surprisingly simple separator structure: in an optimal solution there exists a line from a linear set of candidates that separates one cluster from the rest, enabling efficient evaluation via dynamic minimum enclosing disks/balls. They achieve near-linear time for -MinSumRadius in the plane with expected time and extend to constant dimensions with time; they also obtain a near-quadratic plane algorithm for -MinSumRadius with expected time. The core idea—limiting the search to a linear set of separators and dynamically maintaining MECs—offers a path toward more scalable exact clustering in geometric settings and highlights separator-based techniques for related problems.

Abstract

Given a set of points in the Euclidean plane, the -MinSumRadius problem asks to cover this point set using 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 , and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the -MinSumRadius problem is that of small . For the -MinSumRadius problem, a near-quadratic time algorithm with expected running time 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 -MinSumRadius that runs in expected time. We generalize this result to any constant dimension , for which we give an time algorithm. Additionally, we give a near-quadratic time algorithm for -MinSumRadius in the plane that runs in expected 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.
Paper Structure (12 sections, 8 theorems, 1 figure, 2 algorithms)

This paper contains 12 sections, 8 theorems, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Near-linear algorithm for 2-MinSumRadius
  4. Algorithm
  5. Algorithm description.
  6. Linear number of cuts
  7. Extension to constant dimensions
  8. Near-quadratic algorithm for 3-MinSumRadius
  9. Algorithm
  10. Algorithm description.
  11. Linear number of cuts
  12. Conclusion

Key Result

Lemma 2

Given a $k$-MinSumRadius instance, there exists an optimal solution with clusters $C_1, \dots, C_k$ such that $\Call{MEB}{C_i} \cap \Call{MEB}{C_j} = \emptyset$ for all distinct $i,j \in [k]$.

Figures (1)

  • Figure 1: The optimal $2$-Center clustering (left) compared to the optimal $2$-MinSumRadius clustering (right) for the same point set. In this example $2$-MinSumRadius clustering better captures the structure of the point set than $2$-Center clustering.

Theorems & Definitions (9)

  • Definition 1: $k$-MinSumRadius
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Theorem 8
  • Lemma 9