Over the past decade, a long line of research has investigated the distributed complexity landscape of locally checkable labeling (LCL) problems on bounded-degree graphs, culminating in an almost-complete classification on general graphs and a complete classification on trees. The latter states that, on bounded-degree trees, any LCL problem has deterministic \emph{worst-case} time complexity $O(1)$, $\Theta(\log^* n)$, $\Theta(\log n)$, or $\Theta(n^{1/k})$ for some positive integer $k$, and all of those complexity classes are nonempty. Moreover, randomness helps only for (some) problems with deterministic worst-case complexity $\Theta(\log n)$, and if randomness helps (asymptotically), then it helps exponentially. In this work, we study how many distributed rounds are needed \emph{on average per node} in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic \emph{node-averaged} complexity landscape for LCL problems. As our main result, we show that every problem with worst-case round complexity $O(\log n)$ has deterministic node-averaged complexity $O(\log^* n)$. We further establish bounds on the node-averaged complexity of problems with worst-case complexity $\Theta(n^{1/k})$: we show that all these problems have node-averaged complexity $\widetilde{\Omega}(n^{1 / (2^k - 1)})$, and that this lower bound is tight for some problems.

翻译：过去十年间，一系列研究工作深入探讨了有界度图上局部可检查标记（LCL）问题的分布式复杂度格局，最终在一般图上实现了近乎完整的分类，并在树上完成了完全分类。后者指出：在有界度树上，任何LCL问题的确定性最坏情况时间复杂度为$O(1)$、$\Theta(\log^* n)$、$\Theta(\log n)$或$\Theta(n^{1/k})$（其中$k$为正整数），且所有这些复杂度类均非空。此外，随机性仅对（某些）确定性最坏情况复杂度为$\Theta(\log n)$的问题有助益，且若随机性（渐近地）有效，则其增益呈指数级。本文研究在树上求解LCL问题所需的**每节点平均**分布式轮数。我们获得了LCL问题确定性**节点平均**复杂度格局的部分分类。作为主要结果，我们证明每个最坏情况轮复杂度为$O(\log n)$的问题，其确定性节点平均复杂度为$O(\log^* n)$。我们还进一步建立了最坏情况复杂度为$\Theta(n^{1/k})$问题的节点平均复杂度界限：证明所有此类问题的节点平均复杂度为$\widetilde{\Omega}(n^{1 / (2^k - 1)})$，且该下界对某些问题是紧的。