Distributed Symmetry-Breaking with Improved Vertex-Averaged Complexity

We study the distributed message-passing model in which a communication network is represented by a graph $G=(V,E)$. Usually, the measure of complexity that is considered in this model is the worst-case complexity, which is the largest number of rounds performed by a vertex $v\in V$. Often this is a reasonable measure, but in some occasions it does not express sufficiently well the actual performance of the algorithm. For example, an execution in which one processor performs $r$ rounds, and all the rest perform significantly less rounds than $r$, has the same running time as an execution in which every processor performs $r$ rounds. On the other hand, the latter execution is less efficient in several respects, such as energy efficiency, task execution efficiency, local-neighborhood efficiency and simulation efficiency. Consequently, a more appropriate measure is required in these cases. Recently, the vertex-averaged complexity was proposed by \cite{Feuilloley2017}, where the running time is the worst-case average of rounds over the number of vertices. Feuilloley \cite{Feuilloley2017} showed an improved vertex-averaged complexity can be obtained for leader election, but not for 3-coloring on rings. However, it remained open whether the vertex-averaged complexity of symmetry-breaking in general graphs can be better than the worst-case complexity. In this paper we devise symmetry-breaking algorithms with significantly improved vertex-averaged complexity for general graphs, as well as specific graph families, some of which have considerably better vertex-averaged complexity than the best-possible worst case complexity. For example, for general graphs,we devise an $O(a^{2})$-vertex-coloring algorithm with vertex-averaged complexity of $O(\log\log n)$, where $a$ is the input graph's arboricity. In the worst-case, this requires $\Omega(\log n)$ rounds \cite{Barenboim2008}.