Improved Streaming Edge Coloring

TL;DR

This work advances streaming edge coloring in the W-streaming model by breaking the quadratic color barrier and achieving $O\big(\Delta^{4/3+\varepsilon}\big)$ colors with near-linear space. It introduces a two-pronged strategy: a randomized scheme with $O\big((\log\Delta)^{O(1/\varepsilon)} n\big)$ space and $O\big((\log\Delta)^{O(1/\varepsilon)}\Delta^{4/3+\varepsilon}\big)$ colors (in expectation), and a derandomized variant built on bipartite expanders from multiplicity codes that attains deterministic colorings with comparable, slightly looser bounds. The approach hinges on reducing to bipartite graphs, partitioning the stream into phases, and using forest-structured color palettes and local counters to control color usage, followed by an explicit derandomization step via expanders to remove randomness. Collectively, the results provide near-optimal color efficiency in the W-streaming setting, with practical implications for memory-constrained graph processing and streaming algorithms.

Abstract

Given a graph, an edge coloring assigns colors to edges so that no pairs of adjacent edges share the same color. We are interested in edge coloring algorithms under the W-streaming model. In this model, the algorithm does not have enough memory to hold the entire graph, so the edges of the input graph are read from a data stream one by one in an unknown order, and the algorithm needs to print a valid edge coloring in an output stream. The performance of the algorithm is measured by the amount of space and the number of different colors it uses. This streaming edge coloring problem has been studied by several works in recent years. When the input graph contains $n$ vertices and has maximum vertex degree $Δ$, it is known that in the W-streaming model, an $O(Δ^2)$-edge coloring can be computed deterministically with $\tilde{O}(n)$ space [Ansari, Saneian, and Zarrabi-Zadeh, 2022], or an $O(Δ^{1.5})$-edge coloring can be computed by a $\tilde{O}(n)$-space randomized algorithm [Behnezhad, Saneian, 2024] [Chechik, Mukhtar, Zhang, 2024]. In this paper, we achieve polynomial improvement over previous results. Specifically, we show how to improve the number of colors to $\tilde{O}(Δ^{4/3+ε})$ using space $\tilde{O}(n)$ deterministically, for any constant $ε> 0$. This is the first deterministic result that bypasses the quadratic bound on the number of colors while using near-linear space.

Figures (1)

• Figure 1: In this picture, it shows an example of a forest $\mathcal{T}_\mathbf{f}$ where the orange nodes are the marked ones, and the blue path is the root-to-leaf path ending at the current input batch $F$. Upon the arrival of a new input batch $F$, we need to update the root-to-leaf tree path and the marked sets accordingly.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Corollary 3.2 in ghosh2024low
• Lemma 3.1: implicit in chechik_et_al:LIPIcs.ICALP.2024.40
• proof : Proof sketch
• Lemma 3.2
• Lemma 3.3
• proof : Proof of \ref{['rand']}
• Lemma 3.4
• proof