Core Concepts
First-Fit coloring algorithm uses at most (1/2 + o(1)) · ln n / ln ln n different colors in expectation to color any forest with n vertices.
Abstract
The paper analyzes the performance of the First-Fit coloring algorithm in the random arrival model for the class of forests.
Key highlights:
First-Fit is a simple online coloring algorithm that assigns the least positive integer color to each vertex that is different from its previously colored neighbors.
In the random arrival model, the presentation order of the vertices is chosen uniformly at random, rather than adversarially.
The authors show that First-Fit uses at most (1/2 + o(1)) · ln n / ln ln n different colors in expectation to color any forest with n vertices.
This is a significant improvement over the worst-case performance of First-Fit in the adversarial model, where it uses Θ(log n) colors.
The authors also construct a family of forests for which First-Fit uses (1/2 - o(1)) · ln n / ln ln n different colors in expectation, establishing the tightness of the upper bound.
The analysis provides insights into the average-case behavior of simple online algorithms and lays the groundwork for studying their performance on other graph classes in the random arrival model.
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