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 $\Delta$ under edge arrivals, we obtain an online $O(\Delta)$-coloring in $\tilde{O}(n\sqrt{\Delta})$ space. This is also the first W-streaming edge-coloring algorithm using $O(\Delta)$ colors (in sublinear memory). All prior works either used linear memory or $\omega(\Delta)$ colors. We also achieve a smooth color-space tradeoff: for any $t=O(\Delta)$, we get an $O(\Delta t (\log^2 \Delta))$-coloring in $\tilde{O}(n\sqrt{\Delta/t})$ space, improving upon the state of the art that used $\tilde{O}(n\Delta/t)$ space for the same number of colors (the $\tilde{O}(.)$ 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.

翻译：针对边染色问题，在线模型与W-流式模型似乎互为正交：前者要求在边插入后立即分配颜色，通常不限制存储空间；后者将内存限制为输入规模的次线性，但允许在边插入后的任意时刻宣布其颜色。我们通过设计小空间的在线边染色算法，旨在兼顾两者的优势。我们在（对抗性的）边到达和顶点到达两种场景下研究该问题。我们的结果显著降低了现有在线算法的内存使用量，同时实现了$O(1)$-竞争比。具体而言，对于边到达场景下最大顶点度为$\Delta$的$n$节点图，我们在$\tilde{O}(n\sqrt{\Delta})$空间内获得了在线$O(\Delta)$-染色。这也是首个使用$O(\Delta)$种颜色（在次线性内存中）的W-流式边染色算法。此前所有工作要么使用线性内存，要么使用$\omega(\Delta)$种颜色。我们还实现了平滑的颜色-空间权衡：对于任意$t=O(\Delta)$，我们在$\tilde{O}(n\sqrt{\Delta/t})$空间内获得$O(\Delta t (\log^2 \Delta))$-染色，相较于现有最优方法在相同颜色数下使用$\tilde{O}(n\Delta/t)$空间（$\tilde{O}(.)$符号隐藏了polylog$(n)$因子）取得了改进。这些改进源于对随机排列的广泛使用，使我们能够避免使用先前已分配的颜色。我们的大多数算法可以解随机化，并扩展到多重图（在该场景下边染色被公认为比简单图困难得多）。