Improved Algorithms for Edge Colouring in the W-Streaming Model

TL;DR

This paper introduces two new algorithms for edge colouring in the W-streaming model, achieving near-optimal colour bounds under different edge arrival assumptions with limited space.

Contribution

It presents the first algorithms for edge colouring in the W-streaming model that work under both random and adversarial edge arrival scenarios.

Findings

First algorithm uses $oldsymbol{ riangle + o( riangle)}$ colours with $oldsymbol{O(n ext{polylog} n)}$ space for random permutations.

Second algorithm uses approximately $oldsymbol{ riangle^2 / s}$ colours with $oldsymbol{ ilde{O}(n s)}$ space for adversarial arrivals.

Algorithms operate efficiently within the space constraints of the W-streaming model.

Abstract

In the W-streaming model, an algorithm is given $O(n \mathrm{polylog} n)$ space and must process a large graph of up to $O(n^2)$ edges. In this short note we give two algorithms for edge colouring under the W-streaming model. For edge colouring in W-streaming, a colour for every edge must be determined by the time all the edges are streamed. Our first algorithm uses $\Delta + o(\Delta)$ colours in $O(n \log^2 n)$ space when the edges arrive according to a uniformly random permutation. The second algorithm uses $(1 + o(1))\Delta^2 / s$ colours in $\tilde{O}(n s)$ space when edges arrival adversarially.