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

TL;DR

This paper introduces a simple and efficient dynamic coloring algorithm for sparse graphs that guarantees worst-case polylogarithmic update and query times, improving upon previous amortized results.

Contribution

The paper presents a new simple algorithm for O(α)-implicit coloring with worst-case polylogarithmic update times, strengthening previous amortized guarantees.

Findings

Achieves O(α)-implicit coloring with poly(log n) worst-case update times.

Simplifies the algorithm compared to prior work.

Improves the theoretical guarantees for dynamic graph coloring.

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).