Improved Analysis of Online Balanced Clustering

TL;DR

This paper improves the competitive ratio for online balanced clustering by modifying a known algorithm, making it more practical for large-scale data center applications with many clusters.

Contribution

It introduces a simple modification to the existing algorithm, achieving a nearly linear dependency on the number of clusters, and employs Graver bases for technical analysis.

Findings

Achieved a $( ext{linear in } ext{number of clusters})$-competitive algorithm.

Used Graver bases to analyze solutions of linear integer equations.

Enhanced practical applicability for large data centers.

Abstract

In the online balanced graph repartitioning problem, one has to maintain a clustering of $n$ nodes into $\ell$ clusters, each having $k = n / \ell$ nodes. During runtime, an online algorithm is given a stream of communication requests between pairs of nodes: an inter-cluster communication costs one unit, while the intra-cluster communication is free. An algorithm can change the clustering, paying unit cost for each moved node. This natural problem admits a simple $O(\ell^2 \cdot k^2)$-competitive algorithm COMP, whose performance is far apart from the best known lower bound of $\Omega(\ell \cdot k)$. One of open questions is whether the dependency on $\ell$ can be made linear; this question is of practical importance as in the typical datacenter application where virtual machines are clustered on physical servers, $\ell$ is of several orders of magnitude larger than $k$. We answer this question affirmatively, proving that a simple modification of COMP is $(\ell \cdot 2^{O(k)})$-competitive. On the technical level, we achieve our bound by translating the problem to a system of linear integer equations and using Graver bases to show the existence of a ``small'' solution.