We consider graph coloring and related problems in the distributed message-passing model.
Locally-iterative algorithms
are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its
1-hop-neighborhood
. In STOC’93 Szegedy and Vishwanathan showed that any locally-iterative Δ + 1-coloring algorithm requires Ω (Δ log Δ + log
*
n
) rounds, unless there exists “a very special type of coloring that can be very efficiently reduced” [
44
]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [
2
,
3
,
19
,
32
]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ + 1-coloring algorithm with running time
O
(Δ + log
*
n
), i.e.,
below
Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results:
We obtain self-stabilizing distributed algorithms for Δ + 1-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set, and maximal matching with
O
(Δ + log
*
n
) time. This significantly improves previously known results that have
O(n)
or larger running times [
23
].
We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with
O
(Δ + log
*
n
) time and
O
(Δ)-edge-coloring in the Bit-Round model with
O
(Δ + log
n
) time. The factors of log
*
n
and log
n
are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models.
We obtain an arbdefective coloring algorithm with running time
O
(√ Δ + log
*
n
). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ε)Δ-coloring within
O
(√ Δ + log
*
n
) time and Δ + 1-coloring within
O
(√ Δ log Δ log
*
Δ + log
*
n
) time. This improves the recent state-of-the-art bounds of Barenboim from PODC’15 [
2
] and Fraigniaud et al. from FOCS’16 [
19
] by polylogarithmic factors.
Our algorithms are applicable to the SET-LOCAL model [
25
] (also known as the weak LOCAL model). In this model a relatively strong lower bound of Ω (Δ
1/3
) is known for Δ + 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [
25
] only Linial’s
O
(Δ
2
)-time algorithm and Kuhn-Wattenhofer
O
(Δ log Δ)-time algorithms are shown to work in it.) We obtain the first linear-in-Δ Δ + 1-coloring algorithms that work also in this model.
Deterministic Distributed (Delta + o(Delta))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity
