Speeding Up Constrained $k$-Means Through 2-Means

TL;DR

This paper introduces a faster algorithm for the constrained 2-means problem that produces approximate solutions efficiently, and demonstrates how to improve existing schemes for constrained k-means by leveraging this method.

Contribution

The paper presents a new $O(dn+d({1ackslash ext{epsilon}})^{O(1/ackslashtext{epsilon})} ext{log} n)$ time algorithm for constrained 2-means and shows how to accelerate existing constrained k-means approximation schemes.

Findings

The new algorithm achieves a $(1+ ext{epsilon})$-approximation efficiently.

Existing k-means schemes can be significantly sped up using the proposed method.

The approach reduces the time complexity of previous algorithms for constrained k-means.

Abstract

For the constrained 2-means problem, we present a $O\left(dn+d({1\over\epsilon})^{O({1\over \epsilon})}\log n\right)$ time algorithm. It generates a collection $U$ of approximate center pairs $(c_1, c_2)$ such that one of pairs in $U$ can induce a $(1+\epsilon)$-approximation for the problem. The existing approximation scheme for the constrained 2-means problem takes $O(({1\over\epsilon})^{O({1\over \epsilon})}dn)$ time, and the existing approximation scheme for the constrained $k$-means problem takes $O(({k\over\epsilon})^{O({k\over \epsilon})}dn)$ time. Using the method developed in this paper, we point out that every existing approximating scheme for the constrained $k$-means so far with time $C(k, n, d, \epsilon)$ can be transformed to a new approximation scheme with time complexity ${C(k, n, d, \epsilon)/ k^{\Omega({1\over\epsilon})}}$.