Small Space Representations for Metric Min-Sum k-Clustering and Their Applications

Abstract

The min-sum k -clustering problem is to partition a metric space (P,d) into k clusters $C_1, \ldots,C_k \subseteq P$ such that $\sum_{i=1}^k \sum_{p,q \in C_i} d(p,q)$ is minimized. We show the first efficient construction of a coreset for this problem. Our coreset construction is based on a new adaptive sampling algorithm. Using our coresets we obtain three main algorithmic results.

The first result is a sublinear time (4 + ε)-approximation algorithm for the min-sum k-clustering problem in metric spaces. The running time of this algorithm is $\widetilde{O}(n)$ for any constant k and ε, and it is o(n ²) for all k = o(logn/loglogn). Since the description size of the input is Θ(n ²), this is sublinear in the input size.

Our second result is the first pass-efficient data streaming algorithm for min-sum k-clustering in the distance oracle model, i.e., an algorithm that uses ${\mathit{poly}}(\log n, k)$ space and makes 2 passes over the input point set arriving as a data stream.

Our third result is a sublinear-time polylogarithmic-factor- approximation algorithm for the min-sum k-clustering problem for arbitrary values of k.

To develop the coresets, we introduce the concept of α-preserving metric embeddings. Such an embedding satisfies properties that (a the distance between any pair of points does not decrease, and (b) the cost of an optimal solution for the considered problem on input (P,d′) is within a constant factor of the optimal solution on input (P,d). In other words, the idea is find a metric embedding into a (structurally simpler) metric space that approximates the original metric up to a factor of α with respect to a certain problem. We believe that this concept is an interesting generalization of coresets.

Research supported in part by NSF ITR grant CCR-0313219, by EPSRC grant EP/D063191/1, and by DFG grant Me 872/8-3.