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Parallel Derandomization for Coloring

Sam Coy, Artur Czumaj, Peter Davies, Gopinath Mishra

TL;DR

A novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space is designed and an application is given by showing that a recent (degree + 1) -list coloring algorithm by Halldorsson et al. (STOC’22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized.

Abstract

Graph coloring problems are among the most fundamental problems in parallel and distributed computing, and have been studied extensively in both settings. In this context, designing efficient deterministic algorithms for these problems has been found particularly challenging. In this work we consider this challenge, and design a novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space. We give an application of this framework by showing that a recent $(degree+1)$-list coloring algorithm by Halldorsson et al. (STOC'22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized. Our algorithm runs in $O(\log \log \log n)$ rounds, which matches the complexity of the state of the art algorithm for the $(Δ+ 1)$-coloring problem.

Parallel Derandomization for Coloring

TL;DR

A novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space is designed and an application is given by showing that a recent (degree + 1) -list coloring algorithm by Halldorsson et al. (STOC’22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized.

Abstract

Graph coloring problems are among the most fundamental problems in parallel and distributed computing, and have been studied extensively in both settings. In this context, designing efficient deterministic algorithms for these problems has been found particularly challenging. In this work we consider this challenge, and design a novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space. We give an application of this framework by showing that a recent -list coloring algorithm by Halldorsson et al. (STOC'22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized. Our algorithm runs in rounds, which matches the complexity of the state of the art algorithm for the -coloring problem.
Paper Structure (19 sections, 16 theorems, 1 equation, 12 algorithms)

This paper contains 19 sections, 16 theorems, 1 equation, 12 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Technical overview
  4. Related work
  5. Preliminaries
  6. Notations, the problem, and the model
  7. An overview of $\mathsf{LOCAL}$ $\mathsf{D1LC}$ algorithm of Halldórsson et al. hknt_local_d1lc
  8. Computing an almost-clique decomposition
  9. Coloring sparse nodes
  10. Coloring dense nodes
  11. Randomized $\mathsf{D1LC}$ Algorithm in $\mathsf{MPC}$
  12. Framework to Derandomize $\mathsf{MPC}$ Algorithms
  13. Normal distributed procedures
  14. PRGs and derandomization
  15. Derandomizing normal distributed procedures
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathcal{\phi} \in (0,1)$ be an arbitrary constant. There exists a deterministic algorithm that, for every $n$-node graph $G=(V,E)$, solves the $\mathsf{D1LC}$ problem using $O(\log\log\log n)$ rounds, in the sublinear local space $\mathsf{MPC}$ model with local space $s\xspace = O(n^{\mathcal{

Theorems & Definitions (33)

  • Theorem 1: $\mathsf{D1LC}$
  • Definition 2: Parameters from hknt_local_d1lc
  • Definition 3: $(\hbox{deg}+1)$-ACD AlonA20hknt_local_d1lc
  • Lemma 4
  • Definition 5
  • Definition 6: Definition 7.1 in Vadhan12
  • Definition 7: PRG, Definition 7.3 in Vadhan12
  • Proposition 8: Proposition 7.8 in Vadhan12
  • Lemma 9: Lemma 35 of CDPcompstab, arXiv version
  • Lemma 10
  • ...and 23 more