Given a set of point P in Rd, a clustering problem is to partition P into k subsets {P1, P2, · · · , Pk} in such a way that a given objective function is minimized. The most studied cost functions for a cluster, μ(Pi), are maximum or average radius of Pi, maximum diameter of Pi, and maximum width of Pi. The overall objective function is ⊕ μ(Pi), where ⊕ is typically the Lp-norm operator. The mo...

We consider a classical $k$-center problem in trees. Let $T$ be tree of $n$ vertices such that every vertex has nonnegative weight. The is to find $k$ centers on the edges t...

In this paper we give a first set of communication lower bounds for distributed clustering problems, in particular, for k-center, k-median and k-means. When the input is distributed across a large number of machines and the number of clusters k is small, our lower bounds match the current best upper bounds up to a logarithmic factor. We have designed a new composition framework in our proofs fo...

Metric clustering is a fundamental primitive in machine learning with several applications for mining massive datasets. An important example of metric the k-center problem. While this problem has been extensively studied distributed settings, all previous algorithms use Ω(k) space per and Ω(n k) total work. In paper, we develop first highly scalable approximation algorithm clustering, O~(n^ε) O...

In this paper we develop streaming algorithms for the diameter problem and the k-center clustering problem in the sliding window model. In this model we are interested in maintaining a solution for the N most recent points of the stream. In the diameter problem we would like to maintain two points whose distance approximates the diameter of the point set in the window. Our algorithm computes a ...

This paper investigates the $p$-center location problem on a network in which vertex weights and distances between vertices are uncertain. The concepts of the $alpha$-$p$-center and the expected $p$-center are introduced. It is shown that the $alpha$-$p$-center and the expected $p$-center models can be transformed into corresponding deterministic models. Finally, linear time algorithms for find...

We study the computational complexity of problems that arise in abstract argumentation in the context of dynamic argumentation, minimal change, and aggregation. In particular, we consider the following problems where always an argumentation framework F and a small positive integer k are given. • The Repair problem asks whether a given set of arguments can be modified into an extension by at mos...

Solving geometric optimization problems over uncertain data have become increasingly important in many applications and have attracted a lot of attentions in recent years. In this paper, we study two important geometric optimization problems, the k-center problem and the j-flat-center problem, over stochastic/uncertain data points in Euclidean spaces. For the stochastic k-center problem, we wou...

In this note we showed that a p(≥ 2)-center location problem in general networks can be transformed to the well known Klee’s measure problem [3]. This resulted in an improved algorithm for the continuous case with running time O(mn2 ∗ n log n). The previous best result for the problem is O(mnα(n) log n) where α(n) is the inverse Ackermann function [9]. When the underlying network is a partial k...