[Paper Review] Simple Graph Coloring Algorithms for Congested Clique and Massively Parallel Computation.

Manuela Fischer, Mohsen Ghaffari+1 more|arXiv (Cornell University)|Aug 25, 2018

Complexity and Algorithms in Graphs45 references3 citations

TL;DR

This paper introduces a simple randomized partitioning technique for graph coloring that achieves optimal $(\Delta+1)$-coloring in both the CONGESTED CLIQUE and Massively Parallel Computation (MPC) models. It reduces round complexity to $O(\log^* \Delta)$ in CONGESTED CLIQUE and achieves the first sublogarithmic-time MPC algorithm with $O(n^\alpha)$ memory per machine and $\widetilde{O}(m)$ total memory.

ABSTRACT

We present a very simple randomized partitioning procedure for graph coloring, which leads to simplification or improvements of some recent distributed and parallel coloring algorithms. In particular, we get a simple $(\Delta+1)$ coloring algorithm with round complexity $O(\log^* \Delta)$ in the CONGESTED CLIQUE model of distributed computing. This matches the bound of Parter and Su [DISC'18], which improved on the $O(\log\log \Delta \log^* \Delta)$-round algorithm of Parter [ICALP'18]. Moreover, the same random partitioning leads to a $(\Delta+1)$ coloring algorithm with round complexity $O(\log^* \Delta+ \sqrt{\log\log n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory, which is the first sublogarithmic-time algorithm in this regime. This algorithm uses a memory of $O(n^{\alpha})$ per machine, for any desirable constant $\alpha>0$, and a total memory of $\widetilde{O}(m)$, where $m$ is the size of the graph.

Motivation & Objective

• To simplify and improve recent distributed and parallel graph coloring algorithms using a novel partitioning approach.
• To achieve optimal $(\Delta+1)$-coloring in the CONGESTED CLIQUE model with near-minimal round complexity.
• To design a sublogarithmic-time coloring algorithm in the MPC model with strongly sublinear memory per machine.
• To provide a unified, simple framework applicable across different parallel and distributed models.

Proposed method

• A randomized partitioning procedure is used to decompose the graph into smaller, manageable subproblems.
• Each vertex is assigned to a random partition based on a shared random seed, enabling load balancing and reducing communication overhead.
• The partitioning allows independent coloring of subgraphs, followed by local coordination to resolve conflicts across partitions.
• The method leverages the $\log^* \Delta$-round complexity of prior algorithms, but achieves it more simply through partitioning.
• It ensures that the total memory usage remains $\widetilde{O}(m)$ across all machines in the MPC model.
• The approach is general and can be applied with any $\alpha > 0$, enabling $O(n^\alpha)$ memory per machine.

Experimental results

Research questions

• RQ1 Can a simple randomized partitioning technique achieve optimal $(\Delta+1)$-coloring in the CONGESTED CLIQUE model with $O(\log^* \Delta)$ rounds?
• RQ2 Is it possible to design a sublogarithmic-time MPC algorithm for $(\Delta+1)$-coloring under strongly sublinear memory constraints?
• RQ3 How does the proposed partitioning method compare in efficiency and simplicity to prior complex distributed coloring algorithms?
• RQ4 Can the same technique be adapted to both CONGESTED CLIQUE and MPC models with minimal changes?
• RQ5 What is the memory and round complexity trade-off when using $O(n^\alpha)$ memory per machine in MPC?

Key findings

• The algorithm achieves $O(\log^* \Delta)$-round $(\Delta+1)$-coloring in the CONGESTED CLIQUE model, matching the best-known prior bound.
• The same partitioning method yields the first sublogarithmic-time MPC algorithm for $(\Delta+1)$-coloring with $O(n^\alpha)$ memory per machine.
• The MPC algorithm uses $\widetilde{O}(m)$ total memory, making it efficient and scalable for large graphs.
• The approach simplifies prior complex algorithms while maintaining or improving round complexity.
• The method is robust across models, providing a unified framework for distributed and parallel coloring.
• The algorithm is randomized and efficient, relying only on simple random partitioning and local coordination.