Towards Optimal Deterministic LOCAL Algorithms on Trees
Sebastian Brandt, Ananth Narayanan
TL;DR
This work establishes a constructive bridge between truly local complexity and deterministic tree algorithms in the LOCAL model. By classifying problems into node-labeling and edge-labeling families, the authors present a transformation that converts a given $O(f(\Delta)+\log^* n)$-round algorithm into a tree-specific algorithm running in $O(f(g(n))+\log^* n)$ rounds, where $g(n)$ satisfies $g(n)^{f(g(n))}=n$; a generalized version extends to graphs of bounded arboricity. The approach leverages two decompositions (rake-and-compress for nodes, and a two-parameter edge decomposition for edges) to separate a problem into manageable parts with small diameter or bounded degree, enabling the final runtime to be governed by the truly local complexity. Concrete outcomes include a first strongly sublogarithmic $O(\log^{12/13} n)$-round algorithm for $(\text{edge-degree}+1)$-edge coloring on trees (and related $(2\Delta-1)$-edge coloring bounds) and matching implications for maximal independent set and maximal matching on trees, with extensions to graphs of bounded arboricity. Collectively, the results suggest that tightening truly local complexity bounds could yield near-tight deterministic algorithms on trees, and potentially inform tight bounds on general graphs through a finer problem-by-problem lens.
Abstract
While obtaining optimal algorithms for the most important problems in the LOCAL model has been one of the central goals in the area of distributed algorithms since its infancy, tight complexity bounds are elusive for many problems even when considering \emph{deterministic} complexities on \emph{trees}. We take a step towards remedying this issue by providing a way to relate the complexity of a problem $Π$ on trees to its truly local complexity, which is the (asymptotically) smallest function $f$ such that $Π$ can be solved in $O(f(Δ)+\log^*n)$ rounds. More specifically, we develop a transformation that takes an algorithm $\mathcal A$ for $Π$ with a runtime of $O(f(Δ)+\log^*n)$ rounds as input and transforms it into an $O(f(g(n))+\log^* n)$-round algorithm $\mathcal{A}'$ on trees, where $g$ is the function that satisfies $g(n)^{f(g(n))}=n$. If $f$ is the truly local complexity of $Π$ (i.e., if $\mathcal{A}$ is asymptotically optimal), then $\mathcal{A}'$ is an asymptotically optimal algorithm on trees, conditioned on a natural assumption on the nature of the worst-case instances of $Π$. Our transformation works for any member of a wide class of problems, including the most important symmetry-breaking problems. As an example of our transformation we obtain the first strongly sublogarithmic algorithm for $(\text{edge-degree+1})$-edge coloring (and therefore also $(2Δ-1)$-edge coloring) on trees, exhibiting a runtime of $O(\log^{12/13} n)$ rounds. This breaks through the $Ω(\log n/\log\log n)$-barrier that is a fundamental lower bound for other symmetry-breaking problems such as maximal independent set or maximal matching (that already holds on trees), and proves a separation between these problems and the aforementioned edge coloring problems on trees. We extend a subset of our results to graphs of bounded arboricity.