Attainable accuracy guarantee for the k-medians clustering in [0, 1]

Abstract

We consider the famous k-medians clustering problem in the context of a zero-sum two-player game, which is defined as follows. For given integers $n>1$ and $k>1$, strategy sets of the first and second players consist of n-samples drawn from the unit segment [0, 1] and partitions of the index set $\{1,\ldots , n\}$ into k nonempty subsets (clusters), respectively. As a payoff, we take a loss function of the k-medians clustering evaluated in terms of the sample chosen by the first player and the partition taken by the second one. Actually, the payoff coincides with the sum of distances between points of the sample and the nearest center of a cluster. It is easy to verify that this game has no value. In this paper, for any $n>1$ and $k>1$, we show that $0.5n/(2k-1)$ is an upper bound for the lower value of this game. Furthermore, for any k, we prove attainability of this bound for some ${\bar{n}}={\bar{n}}(k)$ and an arbitrary $n\ge {\bar{n}}$. As a consequence, we show that any n-sample from [0, 1] can be partitioned into k clusters, such that the value of k-medians clustering criterion does not exceed the bound obtained and this bound is tight for sufficiently large n.