We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph \(G=(V,E)\) is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree \(\Delta \) is at least \(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )\). This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also \(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )\)—this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an \(O(\sqrt{\log \log n})\) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is \(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )\), the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matching is \(O(\log ^2\Delta + \log ^* n)\) and the deterministic node-averaged complexity of maximal matching is \(O(\log ^3\Delta + \log ^* n)\). Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be \(\Theta (\log n)\), even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity \(O(\log ^* n)\), while keeping the worst-case complexity in \(O(\log n)\).

我们继续最近开始的关于分布式图算法的分布式节点平均复杂度的工作。在图\(G=(V,E)\)上运行的分布式算法的节点平均复杂度是G的节点V完成计算并提交输出的时间的平均值。我们研究了一些中心分布式对称性破缺问题的节点平均复杂度，并提供了以下结果（其中包括）。作为我们的主要结果，我们表明在最大度数\(\Delta \)的n节点图中计算最大独立集 (MIS) 的随机节点平均复杂度至少为\(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\大)\)。该界限是通过 Kuhn、Moscibroda 和 Wattenhofer [JACM'16] 对著名下界的新颖改编而获得的。作为一个附带结果，我们得到计算树中 MIS 的最坏情况随机轮复杂度也是\(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \ Delta },\sqrt{\frac{\log n}{\log \log n}}\big \}\big )\) —这本质上回答了 Barenboim 和 Elkin 书中的开放问题 11.15，并解决了 MIS 的复杂性在树上达到\(O(\sqrt{\log \log n})\)因素。我们还表明，也许令人惊讶的是，MIS（与 (2, 1) 规则集相同）对 (2, 2) 规则集问题的最小放松将随机节点平均复杂度降低到 O ( 1 ）。对于最大匹配，我们表明，虽然随机节点平均复杂度为\(\Omega \big (\min \big \{\frac{\log \Delta }{\log \log \Delta },\sqrt{\frac {\log n}{\log \log n}}\big \}\big )\)，随机边缘平均复杂度为O (1)。此外，我们证明最大匹配的确定性边缘平均复杂度为\(O(\log ^2\Delta + \log ^* n)\)，最大匹配的确定性节点平均复杂度为\(O(\ log ^3\Delta + \log ^* n)\)。最后，我们考虑计算图的无汇方向的问题。已知问题的确定性最坏情况复杂度为\(\Theta (\log n)\)，即使在有界度图上也是如此。我们证明，可以通过节点平均复杂度\(O(\log ^* n)\)确定性地解决该问题，同时将最坏情况的复杂度保持在\(O(\log n)\)中。