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

AbstractThe min-sumk-clustering problem is to partition a metric space (P,d) into k clusters C1,…,Ck⊆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. With our construction of coresets we obtain two 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 $\widetilde{{\mathcal{O}}}(n)$ for any constant k and ε, and it is o(n2) for all k=o(log n/log log n). Since the full description size of the input is Θ(n2), this is sublinear in the input size. The fastest previously known o(log n)-factor approximation algorithm for k>2 achieved a running time of Ω(nk), and no non-trivial o(n2)-time algorithm was known before.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 poly(log n,k) space and makes 2 passes over the input point set, which arrives in form of a data stream in arbitrary order. It computes an implicit representation of a clustering of (P,d) with cost at most a constant factor larger than that of an optimal partition. Using one further pass, we can assign each point to its corresponding cluster.To develop the coresets, we introduce the concept of α-preserving metric embeddings. Such an embedding satisfies properties that the distance between any pair of points does not decrease and 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 goal is to find a metric embedding into a (structurally simpler) metric space that approximates the original metric up to a factor of αwith respect to a given problem. We believe that this concept is an interesting generalization of coresets.