Distance-2 Coloring in the CONGEST Model

Coloring Fast Without Learning Your Neighbors' Colors

“…As our main result, we give an efficient O(log n)-time randomized algorithm for d2-coloring G with at most ∆ 2 + 1 colors, where ∆ is the maximum degree of G. This improves on a recent O(log ∆ • log n)-time algorithm [23] and it matches the best known bound for ordinary distance-1 (∆+1)-coloring in CONGEST as a function of n alone. We further explore more efficient algorithms when ∆ n. Combining our main method with a range of powerful recent techniques, we obtain an algorithm that runs in time O(log ∆) + 2 O( √ log log n) .…”

“…Our algorithm builds on the approach of [23], which we first summarize. The simple informed color guessing approach -each node tries a random color not used by its d2-neighbors -fails because the nodes do not have the bandwidth to learn those colors.…”

“…The key idea of [23] is to have the colored nodes "help" the live nodes by checking random colors on their neighborhoods. This provides a probabilistic filter that helps reduce the load of the live nodes.…”

“…The nodes are partitioned into a set of sparse nodes-which can be handled by uninformed guesses -and low-diameter clusters of dense nodes. The ACD achieves the same aims as the similarity graphs of [23] that guide the querying and ensure effective filtering, but attain some additional crucial properties such as near-regular high degree. Our extension to ACD is to ensure that all nodes outside clusters have a low degree into the cores of the clusters, strengthening the divide between inside and outside.…”

“…Even the very simple algorithm where each node picks a random available color cannot be efficiently used for d2-coloring as it is in general not possible to keep track of the set of colors chosen by 2-hop neighbors in time o(∆). Recently, Halldórsson, Kuhn and Maus [23] treated d2-coloring in CONGEST and gave a randomized algorithm using ∆ 2 + 1-colors in O(log ∆ • log n) rounds, as well as a deterministic algorithm using (1 + )∆ 2 colors in poly(log n) rounds. Our main approach builds heavily on their framework, while simplifying certain features and strengthening structural properties.…”