Improved Distributed Δ-Coloring

Abstract

<jats:p>We present a randomized distributed algorithm that computes a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Delta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>Δ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-coloring in any non-complete graph with maximum degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Delta \\ge 4$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>Δ</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>4</mml:mn>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\log \\Delta ) + 2^{O(\\sqrt{\\log \\log n})}$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>O</mml:mi>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mi>Δ</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>+</mml:mo>\n <mml:msup>\n <mml:mn>2</mml:mn>\n <mml:mrow>\n <mml:mi>O</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:msqrt>\n <mml:mrow>\n <mml:mo>log</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mi>n</mml:mi>\n </mml:mrow>\n </mml:msqrt>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:msup>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> rounds, as well as a randomized algorithm that computes a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Delta $$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mi>Δ</mml:mi>\n </mml:math></jats:alternatives></jats:inline-formula>-coloring in <jats:inline-formula><jats:alternatives><jats:tex-math>$$O((\\log \\log n)^2)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>O</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:msup>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> rounds when <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Delta \\in [3, O(1)]$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>Δ</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mo>[</mml:mo>\n <mml:mn>3</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mi>O</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>)</mml:mo>\n <mml:mo>]</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. Both these algorithms improve on an <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\log ^3 n / \\log \\Delta )$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>O</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:msup>\n <mml:mo>log</mml:mo>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:mi>n</mml:mi>\n <mml:mo>/</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mi>Δ</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>-round algorithm of Panconesi and Srinivasan (STOC’93), which has remained the state of the art for the past 25\xa0years. Moreover, the latter algorithm gets (exponentially) closer to an <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Omega (\\log \\log n)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>Ω</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mo>log</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> round lower bound of Brandt et al. (STOC’16).</jats:p>