How long it takes for an ordinary node with an ordinary id to output?

Abstract

In the context of distributed synchronous computing, processors perform in rounds, and the time complexity of a distributed algorithm is classically defined as the number of rounds before all computing nodes have output. Hence, this complexity measure captures the running time of the slowest node(s). In this paper, we are interested in the running time of the ordinary nodes, to be compared with the running time of the slowest nodes. The node-averaged time complexity of a distributed algorithm on a given instance is defined as the average, taken over every node of the instance, of the number of rounds before that node outputs. We compare the node-averaged time complexity with the classic one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the former and the later, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two time complexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the node-averaged time complexity is of the same order of magnitude as the "slowest node" time complexity. In addition, in the LOCAL model, the time complexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the ID-averaged time complexity, when the number of rounds is averaged over all possible identity assignments of O(log n)-size identifiers, where n is the size of the network. Our second main result is that the ID-averaged time complexity is essentially the same as the expected time complexity of randomized algorithms (where the expectation is taken over all possible random bits used by the nodes, and the number of rounds is measured for the worst-case identity assignment). Finally, we study the node-averaged IDaveraged time complexity. We show that 3-colouring the n-node ring requires Θ(log * n) rounds if the number of rounds is averaged over the nodes, or if the number of rounds is averaged over the identity assignments. In contrast, we show that 3-colouring the ring requires only O(1) rounds if the number of rounds is averaged over both the nodes and the identity assignments. Laurent Feuilloley 1

Distributed symmetry-breaking with improved vertex-averaged complexity

“…Research on the efficient distributed solution of the above-mentioned central graph-theoretic problems conducted since the 1980's and until a few years ago has focused almost exclusively on the analysis of the time complexity of the developed algorithms in the worst-case scenario. On the other hand, during the past few years, studies presenting new distributed algorithms for the solution of the abovementioned problems with no improvement in the worst-case scenario, but a significant improvement in the average-case scenario have been published ( [26,12]). The analysis of the time complexity of such algorithms in the average-case scenario is based on one of several different models.…”

“…In [12] (and also, in a later published extended version [11]), the model is static, that is, the vertices and edges of the input graph do not change over time. Also, the running time of a vertex for a certain algorithm A is defined in one of two ways, which the author shows to be equivalent.…”

“…Recently, the vertex-averaged complexity was proposed by [12].In this measure, the running time is the worst-case sum of rounds of communication performed by all of the graph's vertices, averaged over the number of vertices. Feuilloley [12] showed that leader-election admits an algorithm with a vertex-averaged complexity significantly better than its worst-case complexity. On the other hand, for O(1)-coloring of rings, the worst-case and vertexaveraged complexities are the same.…”

“…On the other hand, for O(1)-coloring of rings, the worst-case and vertexaveraged complexities are the same. This complexity is O (log * n) [12]. It remained open whether the vertex-averaged complexity of symmetry-breaking in general graphs can be better than the worst-case complexity.…”

“…Consequently, a more appropriate measure is required in these cases. Recently, the vertex-averaged complexity was proposed by [12].…”