Brief Announcement: 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 ε 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 all processors perform the same number of rounds r . 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 \citeFeuilloley2017. In this measure, the running time is the worst-case average of rounds over the number of vertices. Feuilloley \citeFeuilloley2017 showed that leader-election admits an algorithm with significantly better vertex-averaged complexity than worst-case complexity. On the other hand, for $O(1)$-coloring of rings, the worst-case and vertex-averaged complexities are the same. This complexity is O (log* n) [9]. 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 algorithms of ours have significantly 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(loglog n), where the arboricity a is the minimum number of forests that the graph's edges can be partitioned into. In the worst-case, this requires Ω(log n) rounds \citeBarenboim2008.