How fast can you update your MST? (Dynamic algorithms for cluster computing)

TL;DR

This paper presents a nearly optimal distributed algorithm for maintaining a minimum spanning tree in large, dynamic graphs processed by cluster computing models, capable of handling a high rate of updates efficiently.

Contribution

The paper introduces a new algorithm for the $k$-machine and MPC models that processes $O(k)$ updates per $O(1)$ rounds, along with a matching lower bound.

Findings

Algorithm processes $O(k)$ updates per $O(1)$ rounds with high probability.

Lower bound shows processing $k^{1+ ext{epsilon}}$ updates in $O(1)$ rounds is impossible.

Results extend to the Massively Parallel Computation model.

Abstract

Imagine a large graph that is being processed by a cluster of computers, e.g., described by the $k$-machine model or the Massively Parallel Computation Model. The graph, however, is not static; instead it is receiving a constant stream of updates. How fast can the cluster process the stream of updates? The fundamental question we want to ask in this paper is whether we can update the graph fast enough to keep up with the stream. We focus specifically on the problem of maintaining a minimum spanning tree (MST), and we give an algorithm for the $k$-machine model that can process $O(k)$ graph updates per $O(1)$ rounds with high probability. (And these results carry over to the Massively Parallel Computation (MPC) model.) We also show a lower bound, i.e., it is impossible to process $k^{1+\epsilon}$ updates in $O(1)$ rounds. Thus we provide a nearly tight answer to the question of how fast a cluster can respond to a stream of graph modifications while maintaining an MST.