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Approximating k-Center via Farthest-First on $δ$-Covers

Jason R. Wilson

Abstract

The farthest-first traversal of Gonzalez is a classical $2$-approximation algorithm for solving the $k$-center problem, but its sequential nature makes it difficult to scale to very large datasets. In this work we study the effect of running farthest-first on a $δ$-cover of the dataset rather than on the full set of points. A $δ$-cover provides a compact summary of the data in which every point lies within distance $δ$ of some selected center. We prove that if farthest-first is applied to a $δ$-cover, the resulting $k$-center radius is at most twice the optimal radius plus $δ$. In our experiments on large high-dimensional datasets, we show that restricting the input to a $δ$-cover dramatically reduces the running time of the farthest-first traversal while only modestly increasing the $k$-center radius.

Approximating k-Center via Farthest-First on $δ$-Covers

Abstract

The farthest-first traversal of Gonzalez is a classical -approximation algorithm for solving the -center problem, but its sequential nature makes it difficult to scale to very large datasets. In this work we study the effect of running farthest-first on a -cover of the dataset rather than on the full set of points. A -cover provides a compact summary of the data in which every point lies within distance of some selected center. We prove that if farthest-first is applied to a -cover, the resulting -center radius is at most twice the optimal radius plus . In our experiments on large high-dimensional datasets, we show that restricting the input to a -cover dramatically reduces the running time of the farthest-first traversal while only modestly increasing the -center radius.
Paper Structure (13 sections, 5 theorems, 44 equations, 2 algorithms)

This paper contains 13 sections, 5 theorems, 44 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notation
  5. Farthest-First Traversal
  6. Classical Approximation Guarantee
  7. Restricting Farthest-First to Geometric Covers
  8. A Failed Attempt
  9. The Lifting Lemma
  10. A Second Failed Attempt
  11. The Main Result
  12. Empirical Illustration
  13. Conclusion

Key Result

Theorem 1

For every execution of Algorithm 1 on $P$,

Theorems & Definitions (11)

  • Theorem 1: Gonzalez, 1985
  • proof
  • Definition 1: $\delta$-Cover
  • Proposition 1
  • proof
  • Lemma 1: Lifting Lemma
  • proof
  • Lemma 2: Subset Lemma
  • proof
  • Theorem 2
  • ...and 1 more