On the Node-Averaged Complexity of Locally Checkable Problems on Trees
Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Gustav Schmid
TL;DR
This work investigates the node-averaged complexity of locally checkable labeling problems on bounded-degree trees, revealing that for problems with worst-case deterministic time $O(\log n)$, the node-averaged time collapses to $O(\log^* n)$ deterministically and to $O(1)$ with randomness. It introduces and exploits a rake-and-compress decomposition to distribute labeling work, enabling substantial speedups in the average case while preserving worst-case guarantees. For the polynomial regime $\Theta(n^{1/k})$, the authors prove a matching set of lower bounds, show near-optimal node-averaged upper bounds (e.g., $O(n^{1/(2^k-1)})$ for hierarchical $2\frac{1}{2}$-coloring), and establish a $\Omega(n^{1/(2^k-1)}/\log n)$ randomized lower bound that matches up to a polylog factor. Overall, the results significantly advance understanding of how node-averaged complexity behaves in LCLs on trees, highlighting strong randomization gains and tight bounds in polynomial regimes, with implications for energy-efficient or low-overhead distributed computation.
Abstract
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 $O(1)$, $Θ(\log^* n)$, $Θ(\log n)$, or $Θ(n^{1/k})$ 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 $Θ(\log n)$, 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 $Θ(n^{1/k})$: we show that all these problems have node-averaged complexity $\widetildeΩ(n^{1 / (2^k - 1)})$, 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.