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 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 on average per node in order to solve an LCL problem on trees. We obtain a partial classification of the deterministic 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 $O(\log^* n)$. Then we show how using randomization we can speed this up and show that every problem with worst case round complexity $O(\log n)$ has randomized node-averaged complexity $O(1)$. 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. The lower bound holds even for the randomized case and the upper bound is a deterministic algorithm.

翻译：过去十年间，一系列研究深入探索了有界度图上局部可检验标记（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。后者表明：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（部分）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（渐进地）奏效，则其帮助呈指数级增长。本研究聚焦于：在树上求解LCL问题时，每个节点平均需要多少分布式轮次。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：任何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$O(\log^* n)$。进而，我们展示如何借助随机化加速这一过程：任何最坏情况轮复杂度为$O(\log n)$的问题，其随机化节点平均复杂度为$O(1)$。我们进一步建立了最坏情况复杂度为$\Theta(n^{1/k})$的问题的节点平均复杂度界限：证明所有这些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。该下界对随机化情形同样成立，而上界则来自确定性算法。