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 timecomplexity 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 nodeaveraged timecomplexity 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 output. We compare the nodeaveraged timecomplexity with the classical one in the standard LOCAL model for distributed network computing. We show that there can be an exponential gap between the nodeaveraged timecomplexity and the classical timecomplexity, as witnessed by, e.g., leader election. Our first main result is a positive one, stating that, in fact, the two timecomplexities behave the same for a large class of problems on very sparse graphs. In particular, we show that, for LCL problems on cycles, the nodeaveraged time complexity is of the same order of magnitude as the slowest node timecomplexity. In addition, in the LOCAL model, the timecomplexity is computed as a worst case over all possible identity assignments to the nodes of the network. In this paper, we also investigate the IDaveraged timecomplexity, when the number of rounds is averaged over all possible identity assignments. Our second main result is that the IDaveraged timecomplexity is essentially the same as the expected timecomplexity 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 worstcase identity assignment). Finally, we study the nodeaveraged IDaveraged timecomplexity.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 arXiv:
 arXiv:1704.05739
 Bibcode:
 2017arXiv170405739F
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing
 EPrint:
 (Submitted) Journal version