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

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