Classic symmetry-breaking problems on graphs have gained a lot of attention in models of modern parallel computation. The Adaptive Massively Parallel Computation (AMPC) is a model that captures the central challenges in data center computations. Chang et al. [PODC'2019] gave an extremely fast, constant time, algorithm for the $(\Delta + 1)$-coloring problem, where $\Delta$ is the maximum degree of an input graph of $n$ nodes. The algorithm works in the most restrictive low-space setting, where each machine has $n^{\delta}$ local space for a constant $0 < \delta < 1$. In this work, we study the vertex-coloring problem in sparse graphs parameterized by their arboricity $\alpha$, a standard measure for sparsity. We give deterministic algorithms that in constant, or almost constant, time give $\text{poly} ~\alpha$ and $O(\alpha)$-colorings, where $\alpha$ can be arbitrarily smaller than $\Delta$. A strong and standard approach to compute arboricity-dependent colorings is through the Nash-Williams forest decomposition, which gives rise to an (acyclic) orientation of the edges such that each node has a small out-degree. Our main technical contribution is giving efficient deterministic algorithms to compute these orientations and showing how to leverage them to find colorings in low-space AMPC. A key technical challenge is that the color of a node may depend on almost all of the other nodes in the graph and these dependencies cannot be stored on a single machine. Nevertheless, our novel and careful exploration technique yields the orientation, and the arboricity-dependent coloring, with a sublinear number of adaptive queries per node.

翻译：图上的经典对称性破缺问题在现代并行计算模型中备受关注。自适应大规模并行计算（AMPC）是一种捕捉数据中心计算核心挑战的模型。Chang等人[PODC'2019]针对$(\Delta + 1)$-着色问题提出了一种极快的常数时间算法，其中$\Delta$为含$n$个节点的输入图的最大度数。该算法适用于最具限制性的低空间设置，即每台机器拥有$n^{\delta}$的本地空间，$\delta$为常数且满足$0 < \delta < 1$。在本研究中，我们探讨了以树状度$\alpha$（一种标准的稀疏度度量）参数化的稀疏图中的顶点着色问题。我们提出了确定性算法，能在常数时间或近似常数时间内给出$\text{poly} ~\alpha$和$O(\alpha)$-着色，其中$\alpha$可能远小于$\Delta$。计算依赖树状度的着色的经典方法是通过Nash-Williams森林分解，该分解生成边的（无环）定向，使得每个节点的出度较小。我们的主要技术贡献在于：给出了计算这些定向的高效确定性算法，并展示了如何在低空间AMPC中利用它们寻找着色方案。一个关键的技术挑战是节点的颜色可能依赖于图中几乎所有其他节点，而这些依赖关系无法存储于单台机器上。尽管如此，我们新颖且细致的探索技术能够在每个节点仅需对数级自适应查询的情况下，得到定向及依赖树状度的着色方案。