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

For the constrained 2-means problem, we present a $O\left(dn+d({1\overε})^{O({1\over ε})}\log n\right)$ 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 $(1+ε)$-approximation for the problem. The existing approximation scheme for the constrained 2-means problem takes $O(({1\overε})^{O({1\over ε})}dn)$ time, and the existing approximation scheme for the constrained $k$-means problem takes $O(({k\overε})^{O({k\over ε})}dn)$ 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 $C(k, n, d, ε)$ can be transformed to a new approximation scheme with time complexity ${C(k, n, d, ε)/ k^{Ω({1\overε})}}$.