The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to understand the complexity landscape of locally checkable labelings (LCLs) on families of bounded-degree graphs. Particularly interesting in this context is the family of bounded-degree trees as there, for the worst-case complexity, we know a complete characterization of the possible complexities and structures of LCL problems. A first step for the node-averaged complexity case has been achieved recently [DISC '23], where the authors in particular showed that in bounded-degree trees, there is a large complexity gap: There are no LCL problems with a deterministic node-averaged complexity between $\omega(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{\Omega(1)}$. In this work we fill in the remaining gaps and give a complete description of the node-averaged complexity landscape of LCLs on bounded-degree trees. Our contributions are threefold. - On bounded-degree trees, there is no LCL with a node-averaged complexity between $\omega(1)$ and $(\log^*n)^{o(1)}$. - For any constants $0<r_1 < r_2 \leq 1$ and $\varepsilon>0$, there exists a constant $c$ and an LCL problem with node-averaged complexity between $\Omega((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$. - For any constants $0<\alpha\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $\Theta(n^x)$ for some $x\in [\alpha, \alpha+\varepsilon]$.

翻译：节点平均复杂度衡量了在LOCAL模型中，图节点为解决问题平均需要进行的轮次。针对这一新的复杂度度量，一个具有挑战性的研究方向是理解有界度图族上局部可检查标号问题的复杂度全景。特别有趣的是有界度树族，因为对于最坏情况复杂度，我们已经知道LCL问题可能具有的复杂度与结构的完整刻画。最近的工作[DISC '23]迈出了节点平均复杂度情况的第一步，作者特别证明了在有界度树中，存在一个巨大的复杂度间隙：不存在确定性节点平均复杂度在$\omega(\log^* n)$与$n^{o(1)}$之间的LCL问题。对于随机化算法，他们甚至证明了节点平均复杂度要么是$O(1)$，要么是$n^{\Omega(1)}$。在本工作中，我们填补了剩余的空白，给出了有界度树上LCL节点平均复杂度景观的完整描述。我们的贡献有三点：- 在有界度树中，不存在节点平均复杂度介于$\omega(1)$与$(\log^*n)^{o(1)}$之间的LCL问题。- 对于任意常数$0<r_1 < r_2 \leq 1$和$\varepsilon>0$，存在常数$c$及一个LCL问题，其节点平均复杂度介于$\Omega((\log^* n)^c)$与$O((\log^* n)^{c+\varepsilon})$之间。- 对于任意常数$0<\alpha\leq 1/2$和$\varepsilon>0$，存在一个LCL问题，其节点平均复杂度为$\Theta(n^x)$，其中$x\in [\alpha, \alpha+\varepsilon]$。