Locally Checkable Problems in Rooted Trees

Abstract

Consider any locally checkable labeling problem Π in rooted regular trees: there is a finite set of labels Σ, and for each label ∈ Σ we specify what are permitted label combinations of the children for an internal node of label (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set.We show that the distributed computational complexity of any such problem Π falls in one of the following classes: it is (1), Θ(log * ), Θ(log ), or Θ(1) rounds in trees with nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic LOCAL, randomized LOCAL, deterministic CONGEST, and randomized CONGEST model. In particular, we show that randomness does not help in this setting, and the complexity class Θ(log log ) does not exist (while it does exist in the broader setting of general trees).We also show how to systematically determine the complexity class of any such problem Π, i.e., whether Π takes (1), Θ(log * ), Θ(log ), or Θ(1) rounds. While the algorithm may take exponential time in the size of the description of Π, it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.

The Landscape of Distributed Complexities on Trees and Beyond

“…A recent work [8] gave a complete classification of possible complexities of LCLs on rooted regular trees, showing that each LCL on such trees has a complexity of Op1q, Θplog ˚nq, Θplog nq, or Θpn 1{k q for some positive integer k (and all of these complexity classes are nonempty). Moreover, the complexity of each LCL is independent of whether randomization is allowed and whether the LOCAL or the CONGEST model 3 is considered.…”

“…For instance, the asymptotic complexity of problems from a natural subclass of LCLs, called binary labeling problems can be decided efficiently [4]. Moreover, in their work providing the complexity clas-sification of LCLs on regular rooted trees [8], the authors showed that it is decidable into which of the four complexity classes Op1q, Oplog ˚nq, Θplog nq, and n Θp1q a given LCL without inputs falls. The decidability is achieved by defining so-called "certificates" for Oplog nq-round, Oplog ˚nq-round, and constant-round solvability-for each T from tOplog nq, Oplog ˚nq, Op1qu, an LCL problem is T -round solvable if and only if there exists a certificate for T -round solvability for the given LCL problem, and the existence of such a certificate is decidable.…”

“…Let Π be a node-edge-checkable LCL problem and A a randomized algorithm solving Π on F with runtime T pnq and local failure probability at most p ď 1. 8 Let N be the set of all positive integers n satisfying T pnq `2 ď log ∆ n. Then, there exists a randomized algorithm A 1 solving RpRpΠqq on F N with runtime maxt0, T pnq ´1u and local failure probability at most Sp 1{p3∆`3q , where…”

“…The notion of locality is captured by the local complexity of the LOCAL model of distributed computing [36]. In recent years, our understanding of locality has improved dramatically by studying it from a complexity theoretical perspective [41,19,21,17,8]. That is, characterizing the local complexity of all problems from a certain class of problems at once.…”

Online Algorithms with Lookaround

“…They considered bounded-degree graphs and showed that there exist non-trivial problems, such as the weak 2-coloring that can be solved in constant time. Today, LCL problems are well classified with respect to their computational complexity for the special cases of paths [3,4,11,16,35], grids [11], directed and undirected trees [4,5,7,13,15] as well as general graphs [11,35], with only a few unknown gaps [5]. In an attempt to overcome issues with symmetry-breaking in the LOCAL model, the SLOCAL [23] model has been introduced.…”