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Density-Dependent Graph Orientation and Coloring in Scalable MPC

Mohsen Ghaffari, Christoph Grunau

TL;DR

This paper presents massively parallel computation algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density, where α denotes the density of the densest subgraph.

Abstract

This paper presents massively parallel computation (MPC) algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density. Our algorithms run in $poly(\log\log n)$ rounds and compute an orientation of the edges with maximum outdegree $O(α\log\log n)$ as well as a coloring of the vertices with $O(α\log\log n)$ colors. Here, $α$ denotes the density of the densest subgraph. Our algorithm's round complexity is notable because it breaks the $\tildeΘ(\sqrt{\log n})$ barrier, which applied to the previously best known density-dependent orientation algorithm [Ghaffari, Lattanzi, and Mitrovic ICML'19] and is common to many other scalable MPC algorithms.

Density-Dependent Graph Orientation and Coloring in Scalable MPC

TL;DR

This paper presents massively parallel computation algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density, where α denotes the density of the densest subgraph.

Abstract

This paper presents massively parallel computation (MPC) algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density. Our algorithms run in rounds and compute an orientation of the edges with maximum outdegree as well as a coloring of the vertices with colors. Here, denotes the density of the densest subgraph. Our algorithm's round complexity is notable because it breaks the barrier, which applied to the previously best known density-dependent orientation algorithm [Ghaffari, Lattanzi, and Mitrovic ICML'19] and is common to many other scalable MPC algorithms.
Paper Structure (32 sections, 16 theorems, 79 equations, 4 algorithms)

This paper contains 32 sections, 16 theorems, 79 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Models and Basic Definitions
  3. Massively Parallel Computation Model.
  4. LOCAL Model.
  5. Densest Subgraph Density, and Abroricity.
  6. Low Out-degree Orientation.
  7. State of the Art in Scalable MPC and Distributed Algorithms
  8. LOCAL Algorithms.
  9. Scalable MPC Algorithms.
  10. Broader Context---LOCAL vs Scalable MPC, graph exponentiation, and sparisifcation.
  11. Our Results
  12. Discussion.
  13. Technical Overview
  14. Orientation
  15. Coloring.
  16. ...and 17 more sections

Key Result

Theorem 1.1

There is a randomized scalable MPC algorithm that given any undirected graph $G=(V, E)$ with $n=|V|$ and $m=|E|$, runs in $\operatorname{\text{\rm poly}}(\log \log n)$ rounds and computes an orientation of the edges such that each node has outdegree at most $O(\lambda \log\log n)$, with high probabi

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Edge Partitioning
  • proof
  • Lemma 2.2: Vertex Partitioning
  • proof
  • Definition 2.1: Partial Layer Assignment
  • Claim 2.3: Min of two partial layer assignments
  • proof
  • Definition 2.2: Strictly Increasing Paths and Path Counts
  • ...and 43 more