Core Concepts

We develop a general framework for derandomizing randomized LOCAL algorithms in the deterministic low-space Massively Parallel Computation (MPC) model, and apply it to obtain an efficient deterministic MPC algorithm for the Degree+1 List Coloring (D1LC) problem.

Abstract

The paper presents a novel framework for derandomizing randomized algorithms in the Massively Parallel Computation (MPC) model with sublinear local space. The key idea is to decompose the randomized algorithm into short subprocedures with certain useful properties, and then derandomize these subprocedures individually using pseudorandom generators and the method of conditional expectations.
As an application, the authors consider the Degree+1 List Coloring (D1LC) problem, which is a generalization of the well-studied (Δ+1)-coloring problem. They first show how to translate the recent randomized LOCAL algorithm for D1LC by Halldórsson et al. to the MPC setting, obtaining a randomized MPC algorithm that runs in O(log log log n) rounds. Then, they apply their derandomization framework to this randomized MPC algorithm, obtaining a deterministic MPC algorithm for D1LC that also runs in O(log log log n) rounds.
The key technical contribution is the derandomization framework, captured in Theorem 12, which formalizes a set of properties that allow for efficient derandomization in the MPC model. This framework significantly extends prior work on derandomization in MPC, which was either heavily tailored to specific algorithms or could only handle low-degree instances.

Stats

The maximum degree of the input graph is denoted by Δ.
The local space parameter of the MPC model is denoted by φ, where the local space on each machine is O(nφ).

Quotes

"We develop a general derandomization framework, providing a useful tool for translating some class of randomized LOCAL algorithms to deterministic MPC in a black-box manner."
"Our framework allows generic derandomization of a large class of algorithms over a much larger degree range."

Key Insights Distilled From

by Sam Coy,Artu... at **arxiv.org** 04-25-2024

Deeper Inquiries

The derandomization framework developed in this work can be applied to other fundamental graph problems beyond coloring by adapting the techniques to suit the specific requirements of those problems. The key idea is to identify randomized algorithms in the LOCAL model that can be derandomized using pseudorandom generators and conditional expectations. By analyzing the properties of these algorithms and understanding how they can be translated into deterministic procedures in the MPC model, it is possible to extend the framework to address a variety of graph problems.
For example, problems like connectivity, maximal matching, maximal independent set, and various other graph optimization problems can benefit from derandomization techniques similar to those presented in this work. By identifying the key components of randomized algorithms in the LOCAL model and applying the derandomization framework, it is feasible to develop efficient deterministic MPC algorithms for a wide range of graph problems.

Yes, the techniques used in this work can be extended to obtain deterministic MPC algorithms for other variants of graph coloring, such as edge coloring or list edge coloring. The framework for derandomizing algorithms in the MPC model can be adapted to suit the specific requirements of these variants by considering the unique characteristics of edge coloring and list edge coloring problems.
By analyzing the randomized algorithms designed for edge coloring or list edge coloring in the LOCAL model, it is possible to identify the components that can be derandomized using pseudorandom generators and conditional expectations. The key lies in understanding how to translate these algorithms into deterministic procedures that can efficiently solve edge coloring or list edge coloring problems in the MPC model while maintaining the desired complexity bounds.

The 1-vs-2-cycles conjecture used in the paper has limitations in that it serves as a conditional assumption for the complexity of certain graph problems in the sublinear local space MPC model. While the conjecture provides insights into the relationship between the complexity of problems in the LOCAL model and the MPC model, it is not a definitive proof of lower bounds or limitations in the MPC model.
To bypass the 1-vs-2-cycles conjecture and potentially obtain even faster deterministic MPC algorithms for coloring problems, researchers can explore alternative approaches and techniques. This may involve developing new derandomization frameworks that do not rely on specific conjectures or assumptions, or finding innovative ways to optimize existing algorithms for coloring problems in the MPC model.
Additionally, further research could focus on refining the analysis of deterministic algorithms for coloring problems in the MPC model, exploring different strategies for reducing the complexity and improving the efficiency of these algorithms without being constrained by the assumptions of the 1-vs-2-cycles conjecture.

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