The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the $$\mathsf {LOCAL}$$ LOCAL model and locally checkable problems ( $$\mathsf {LCL}$$ LCL s) in bounded-degree graphs, the following picture emerges: There are lots of problems with time complexities of $$\varTheta (\log ^* n)$$ Θ ( log ∗ n ) or $$\varTheta (\log n)$$ Θ ( log n ) . It is not possible to have a problem with complexity between $$\omega (\log ^* n)$$ ω ( log ∗ n ) and $$o(\log n)$$ o ( log n ) . In general graphs , we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\log n)$$ ω ( log n ) and $$n^{o(1)}$$ n o ( 1 ) . In trees , problems with such complexities do not exist. However, the high end of the complexity spectrum was left open by prior work. In general graphs there are $$\mathsf {LCL}$$ LCL problems with complexities of the form $$\varTheta (n^\alpha )$$ Θ ( n α ) for any rational $$0 < \alpha \le 1/2$$ 0 < α ≤ 1 / 2 , while for trees only complexities of the form $$\varTheta (n^{1/k})$$ Θ ( n 1 / k ) are known. No $$\mathsf {LCL}$$ LCL problem with complexity between $$\omega (\sqrt{n})$$ ω ( n ) and o ( n ) is known, and neither are there results that would show that such problems do not exist. We show that: In general graphs , we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\sqrt{n})$$ ω ( n ) and o ( n ). In trees , problems with such complexities do not exist. Put otherwise, we show that any $$\mathsf {LCL}$$ LCL with a complexity o ( n ) can be solved in time $$O(\sqrt{n})$$ O ( n ) in trees, while the same is not true in general graphs.

中文翻译：

---

LOCAL 模型中的几乎全局问题

对于子多项式复杂性，分布式时间复杂度的前景如今已得到很好的理解。当我们查看 $$\mathsf {LOCAL}$$ LOCAL 模型中的确定性算法和有界度图中的局部可检查问题 ( $$\mathsf {LCL}$$ LCL s) 时，会出现下图：时间复杂度为 $$\varTheta (\log ^* n)$$ Θ ( log ∗ n ) 或 $$\varTheta (\log n)$$ Θ ( log n ) 的问题。在 $$\omega (\log ^* n)$$ ω ( log ∗ n ) 和 $$o(\log n)$$ o ( log n ) 之间不可能存在复杂性问题。在一般图中，我们可以构造 $$\mathsf {LCL}$$ LCL 问题在 $$\omega (\log n)$$ ω ( log n ) 和 $$n^{o(1)} 之间具有无限复杂度$$ 没有（1）。在树中，不存在具有这种复杂性的问题。然而，复杂范围的高端被先前的工作留下来了。在一般图形中，对于任何有理数 $$0 < \alpha \le 1/ 存在具有 $$\varTheta (n^\alpha )$$ Θ ( n α ) 形式复杂性的 $$\mathsf {LCL}$$ LCL 问题2$$ 0 < α ≤ 1 / 2 ，而对于树，只有 $$\varTheta (n^{1/k})$$ Θ ( n 1 / k ) 形式的复杂性是已知的。没有已知复杂度介于 $$\omega (\sqrt{n})$$ ω ( n ) 和 o ( n ) 之间的 $$\mathsf {LCL}$$ LCL 问题，也没有结果表明这样的问题不存在。我们证明： 在一般图中，我们可以构造 $$\mathsf {LCL}$$ LCL 问题，在 $$\omega (\sqrt{n})$$ ω ( n ) 和 o ( n ) 之间具有无限多的复杂性。在树中，不存在具有这种复杂性的问题。否则，