Distributed Testing of Distance-k Colorings

Abstract

We study the distributed decision problem related to checking distance-k coloring, defined as color assignments to the nodes such that every pair of vertices at distance at most k must receive distinct colors. While checking the validity of a distance-k coloring only requires $\lceil k/2\rceil$ rounds in the Local model, and a single round in the Congest model when $k\le 2$, the task is extremely costly for higher k’s in Congest—there is a lower bound of $\varOmega (\varDelta ^{k/2})$ rounds in graphs with maximum degree $\varDelta$. We therefore explore the ability of checking distance-k coloring via distributed property testing. We consider several farness criteria for measuring the distance to a valid coloring, and we derive upper and lower bounds for each of them. In particular, we show that for one natural farness measure, significantly better algorithms are possible for testing distance-3 coloring than for testing distance-k coloring for $k \ge 4$.

Pierre Fraigniaud is partially supported by ANR Projects DESCARTES, QuDATA, and FREDDA; Magnús M. Halldórsson and Alexandre Nolin are partially supported by Icelandic Research Foundation grant 174484-051.

A Verifying Distance-k Colorings in Bounded-Degree Graphs

1.1 A.1 A matching lower bound for the natural algorithm

In a graph of maximum degree $\varDelta$, the nodes can learn their distance-$\lceil k / 2 \rceil$ neighborhood in $O\left( \varDelta ^{\lceil k/2\rceil -1}\right)$ rounds in Congest. In particular, an invalid distance-k coloring can be detected with this number of rounds in Congest, since two nodes of distance at most k are both within a distance $\lceil k / 2 \rceil$ of some node. This protocol is actually close to optimal, as our next theorem shows.

Theorem 7

For $k \ge 3$, the verification of a distance-k coloring requires $\widetilde{\varOmega }\left( \varDelta ^{\lceil k/2\rceil -1}\right)$ rounds in the Congest model.

Fig. 7.

The graph we use for our lower bound. It consists of 2 complete $(\varDelta -1)$-ary trees of depth $\lceil k/2\rceil -1$ linked at their roots.

Proof sketch. The proof again relies on embedding a Set Disjointness instance in a graph (see Fig. 7). Here, a Set Disjointness instance with sets of size up to $\varTheta (\varDelta -^{\lceil k/2\rceil -1})$ and no promise on the intersection can be embedded, with a single edge connecting Alice’s and Bob’s parts of the graph.