Simple, Deterministic, Constant-Round Coloring in the Congested Clique

We settle the complexity of the (Δ + 1)-coloring and (Δ + 1)-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round (Δ + 1)-list coloring algorithm due to Chang et al. (PODC'19), and significantly improves upon the state-of-the-art O(log Δ)-round deterministic (Δ + 1)-coloring bound of Parter (ICALP'18).

A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithms for this problem are based on the quite involved local coloring algorithm of Chang et al. (STOC'18), our algorithm can be described in just a few lines. At a high level, it applies a careful derandomization of a recursive procedure which partitions the nodes and their respective palettes into separate bins. We show that after O(1) recursion steps, the remaining uncolored subgraph within each bin has linear size, and thus can be solved locally by collecting it to a single node. This algorithm can also be implemented in the Massively Parallel Computation (MPC) model provided that each machine has linear (in n, the number of nodes in the input graph) space.

We also show an extension of our algorithm to the MPC regime in which machines have sublinear space: we present the first deterministic (Δ + 1)-list coloring algorithm designed for sublinear-space MPC, which runs in O(log Δ + log log n) rounds.