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 for $O(\Delta)$-coloring 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 $O(\Delta (\log \Delta)^2 t)$-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 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：对于边着色问题，在线模型和W-流式模型似乎存在某种正交性：前者要求边在插入后立即分配颜色，通常不限制内存空间；后者虽将内存限制为输入规模的次线性，但允许边在插入后的任意时刻获得颜色。我们致力于通过设计小空间在线边着色算法，兼顾两者优势。我们分别研究了（对抗性）边到达和顶点到达两种情形下的问题。我们的结果显著降低了先前在线算法的内存开销，同时实现了$O(1)$竞争比。具体而言，对于边到达下最大顶点度为$\Delta$的$n$节点图，我们以$\tilde{O}(n\sqrt{\Delta})$空间实现了在线$O(\Delta)$-着色。这也是首个在次线性内存中实现$O(\Delta)$-着色的W-流式边着色算法。此前所有工作要么采用线性内存，要么使用$\omega(\Delta)$种颜色。我们还实现了颜色与空间的平滑权衡：对于任意$t=O(\Delta)$，我们以$\tilde{O}(n\sqrt{\Delta/t})$空间得到$O(\Delta (\log \Delta)^2 t)$-着色，而同等颜色数量下先前最优方案需使用$\tilde{O}(n\Delta/t)$空间。这些改进源于对随机排列的充分运用，使我们能够避开已用过的颜色。我们的大部分算法可去随机化并推广到多重图（已知其边着色问题远比简单图困难）。