Speeding Up Constrained $k$-Means Through 2-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 $${C(k, n, d, \epsilon)/ k<sup>{\Omega({1\over\epsilon})}}$$.