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Dynamic O(arboricity) coloring in polylogarithmic worst-case time

Mohsen Ghaffari, Christoph Grunau

TL;DR

This work presents an extremely simple algorithm that computes an O(α)-implicit coloring with poly(logn) amortized update and query times and shows that the time complexity guarantee can be strengthened from amortized to worst-case.

Abstract

A recent work by Christiansen, Nowicki, and Rotenberg provides dynamic algorithms for coloring sparse graphs, concretely as a function of the arboricity alpha of the input graph. They give two randomized algorithms: O({alpha} log {alpha}) implicit coloring in poly(log n) worst-case update and query times, and O(min{{alpha} log {alpha}, {alpha} log log log n}) implicit coloring in poly(log n) amortized update and query times (against an oblivious adversary). We improve these results in terms of the number of colors and the time guarantee: First, we present an extremely simple algorithm that computes an O({alpha})-implicit coloring with poly(log n) amortized update and query times. Second, and as the main technical contribution of our work, we show that the time complexity guarantee can be strengthened from amortized to worst-case. That is, we give a dynamic algorithm for implicit O({alpha})-coloring with poly(log n) worst-case update and query times (against an oblivious adversary).

Dynamic O(arboricity) coloring in polylogarithmic worst-case time

TL;DR

This work presents an extremely simple algorithm that computes an O(α)-implicit coloring with poly(logn) amortized update and query times and shows that the time complexity guarantee can be strengthened from amortized to worst-case.

Abstract

A recent work by Christiansen, Nowicki, and Rotenberg provides dynamic algorithms for coloring sparse graphs, concretely as a function of the arboricity alpha of the input graph. They give two randomized algorithms: O({alpha} log {alpha}) implicit coloring in poly(log n) worst-case update and query times, and O(min{{alpha} log {alpha}, {alpha} log log log n}) implicit coloring in poly(log n) amortized update and query times (against an oblivious adversary). We improve these results in terms of the number of colors and the time guarantee: First, we present an extremely simple algorithm that computes an O({alpha})-implicit coloring with poly(log n) amortized update and query times. Second, and as the main technical contribution of our work, we show that the time complexity guarantee can be strengthened from amortized to worst-case. That is, we give a dynamic algorithm for implicit O({alpha})-coloring with poly(log n) worst-case update and query times (against an oblivious adversary).

Paper Structure

This paper contains 24 sections, 6 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Background: Arboricity-Dependent Coloring and Dynamic Algorithms
  3. Arboricity-Dependent Coloring
  4. Dynamic Algorithms
  5. Explicit vs. Implicit Outputs
  6. State of the Art
  7. Orientation and Coloring
  8. Direct implications of classic distributed coloring algorithms
  9. Improved Colorings of Christiansen et al. Christiansen23
  10. Our Results
  11. Preliminaries
  12. Our Algorithms
  13. Coloring with Amortized Query Time
  14. Algorithm Framework
  15. How do we color $V^*$?
  16. ...and 9 more sections

Key Result

Theorem 1.1

There is a randomized dynamic algorithm that computes an $O(\alpha)$ implicit coloring in $\operatorname{poly}(\log n)$ worst-case update and query times, against an oblivious adversary.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Christiansen, Holm, Van der Hoog, Rotenberg, Schwiegelshohn christiansen2022adaptive
  • Theorem 2.2: Christiansen, Nowicki, and Rotenberg Christiansen23
  • Lemma 3.1
  • proof
  • proof : Proof of \ref{['thm:amortized']}
  • Lemma 3.2
  • proof
  • proof : Proof of \ref{['thm:main']}