Speeding Up Constrained $k$-Means Through 2-Means
(1808.04062)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 $(c1, c2)$ 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 ${C(k, n, d, \epsilon)/ k{\Omega({1\over\epsilon})}}$.
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