Sam Coy

Sam Coy, Artur Czumaj, Gopinath Mishra

We consider the classic $k$-center problem {in the constant dimensional Euclidean space} under a parallel setting, on the low-local-space Massively Parallel Computation (MPC) model, with local space per machine of ${O}(n^δ)$, where $δ\in (0,1)$ is an arbitrary constant. As a central clustering problem, the $k$-center problem has been studied extensively. Still, until very recently, all parallel MPC algorithms have been requiring $Ω(k)$ or even $Ω(k n^δ)$ local space per machine. While this setting covers the case of small values of $k$, for a large number of clusters these algorithms require large local memory, making them poorly scalable. The case of large $k$, $k \ge Ω(n^δ)$, has been considered recently for the low-local-space MPC model by Bateni et al.\ (2021), who gave an ${O}(\log \log n)$-round MPC algorithm that produces $k(1+o(1))$ centers whose cost has multiplicative approximation of ${O}(\log\log\log n)$. In this paper we extend the algorithm of Bateni et al. and design a low-local-space MPC algorithm that in ${O}(\log\log n)$ rounds returns a clustering with $k(1+o(1))$ clusters that is an ${O}(\log^*n)$-approximation for $k$-center.

Sam Coy, Artur Czumaj, Peter Davies et al.

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node $u$ of degree $d(u)$ is assigned a palette of $d(u)+1$ colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds.