[Paper Review] Low-Memory Algorithms for Online Edge Coloring

Prantar Ghosh, Manuel Stoeckl|arXiv (Cornell University)|Apr 24, 2023

Complexity and Algorithms in Graphs2 citations

TL;DR

This paper presents low-memory online and W-streaming algorithms for edge coloring that achieve an O(∆)-coloring using sublinear space, significantly reducing memory usage compared to prior work. By leveraging random permutations and dynamic color pool management, the algorithms enable O(∆)-competitive coloring in Õ(n√∆) space for edge arrivals and semi-streaming space for vertex arrivals, with smooth tradeoffs between memory and color count.

ABSTRACT

For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to sublinear in the input size but allows an edge's color to be announced any time after its insertion. We aim for the best of both worlds by designing small-space online algorithms for edge coloring. We study the problem under both (adversarial) edge arrivals and vertex arrivals. Our results significantly improve upon the memory used by prior online algorithms while achieving an $O(1)$-competitive ratio. In particular, for $n$-node graphs with maximum vertex-degree $Δ$ under edge arrivals, we obtain an online $O(Δ)$-coloring in $ ilde{O}(n\sqrtΔ)$ space. This is also the first W-streaming edge-coloring algorithm using $O(Δ)$ colors (in sublinear memory). All prior works either used linear memory or $ω(Δ)$ colors. We also achieve a smooth color-space tradeoff: for any $t=O(Δ)$, we get an $O(Δt (\log^2 Δ))$-coloring in $ ilde{O}(n\sqrt{Δ/t})$ space, improving upon the state of the art that used $ ilde{O}(nΔ/t)$ space for the same number of colors (the $ ilde{O}(.)$ notation hides polylog$(n)$ factors). The improvements stem from extensive use of random permutations that enable us to avoid previously used colors. Most of our algorithms can be derandomized and extended to multigraphs, where edge coloring is known to be considerably harder than for simple graphs.

Motivation & Objective

• To design online edge coloring algorithms that use sublinear memory while maintaining an O(∆)-competitive ratio.
• To bridge the gap between online and W-streaming models by enabling immediate color assignment with limited memory.
• To improve upon prior work that required linear memory or used ω(∆) colors in sublinear space.
• To generalize the approach to multigraphs and provide deterministic variants.
• To establish smooth space-color tradeoffs for both edge-arrival and vertex-arrival settings.

Proposed method

• Uses random permutations to efficiently sample and avoid previously used colors without storing all prior assignments.
• Maintains dynamic color pools at each vertex, incrementally expanding them using permutations to find available colors.
• Employs a 'try-and-discard' strategy with oracle randomness to avoid explicitly storing all random choices.
• Introduces a novel W-streaming model where colors can be deferred but memory is sublinear, enabling space-efficient processing.
• Derandomizes algorithms using structured randomness and proves correctness for multigraphs.
• Achieves space bounds via careful analysis of path restrictions in switching networks and k-wise independence.

Experimental results

Research questions

• RQ1 Can we achieve O(∆)-coloring in online edge coloring with sublinear memory?
• RQ2 Can we design a W-streaming edge coloring algorithm that uses O(∆) colors and sublinear space?
• RQ3 What is the optimal tradeoff between memory usage and number of colors in online edge coloring?
• RQ4 Can the proposed techniques be extended to multigraphs and made deterministic?
• RQ5 Can we reduce memory usage below linear while maintaining a constant competitive ratio?

Key findings

• Achieves an O(∆)-coloring in Õ(n√∆) space for adversarial edge arrivals, improving upon prior online algorithms that used more memory.
• Introduces the first W-streaming edge coloring algorithm using O(∆) colors in sublinear memory, resolving a key open problem.
• Establishes a smooth space-color tradeoff: O(∆t)-coloring in Õ(n√∆/t) space, improving over the state of the art that used Õ(n∆/t) space for the same color count.
• For vertex-arrival models, reduces space usage to O(n polylog n), achieving semi-streaming bounds in general graphs and bipartite graphs.
• All algorithms can be derandomized and generalized to multigraphs, where edge coloring is more complex.
• Conjectures that randomized algorithms use Õ(n) and Õ(n√∆) space with high probability, though formal proofs remain open.