A network model of barrier synchronization algorithms

We consider two algorithms for the barrier synchronization ofN processes: the Dissemination algorithm⁽²⁾ and Brooks algorithm.^(1,2) Both algorithms comprise a number of binary communications amongst the processes, organized into a sequence of stages. It is shown that Brooks' algorithm⁽¹⁾ requires between LogN(log₂ N) and 2 LogN stages, the lower bound being guaranteed only in the case thatN is a power of 2 (cubic) and the upper bound seemingly needed for most otherN. On the other hand, it is shown⁽²⁾ that the Dissemination algorithm requires only LogN stages regardless ofN, making it apparently superior to the Brooks algorithm. We introduce a network model of local barrier algorithms. Using it we obtain a rigorous correctness proof for local barrier algorithms, and show that the number of stages in the Brooks algorithm is bounded above by LogN+1. The Brooks algorithms is therefore essentially equivalent in time complexity to the Dissemination algorithm. We then address the question of which values ofN admit exactly LogN Brooks stages. We find a sufficient condition, and conjecture that it is also necessary.