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 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 outdegree. 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：经典图上的对称性破坏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$\Delta$是包含$n$个节点的输入图的最大度数。该算法在最严格的小空间设置下运行，即每台机器仅拥有$n^{\delta}$的本地空间，其中常数$0 < \delta < 1$。本研究关注稀疏图中以树度$\alpha$参数化的顶点着色问题（$\alpha$是稀疏性的标准度量）。我们提出确定性算法，能在常数或近似常数时间内生成$\text{poly}(\alpha)$和$O(\alpha)$-着色，其中$\alpha$可能远小于$\Delta$。计算依赖树度的着色的一种经典标准方法是基于Nash-Williams森林分解，该分解产生边的（无环）定向，使得每个节点具有较小的出度。我们的主要技术贡献在于给出了高效确定性算法来计算这些定向，并展示了如何利用它们在低空间AMPC中寻找着色。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在单台机器上。尽管如此，我们新颖且细致的探索技术为每个节点生成了具有亚线性自适应查询次数的定向及依赖树度的着色。