The impact of randomization in smoothing networks

We revisit randomized smoothing networks (Herlihy and Tirthapura in J Parallel Distrib Comput 66(5): 626-632, 2006), which are made up of balancers and wires. We assume that balancers are oriented independently and uniformly at random. Tokens arrive arbitrarily onw input wires and propagate asynchronously through the network; each token is served on the output wire it arrives at. The smoothness is the maximum discrepancy among the numbers of tokens arriving at the w output wires. We present a collection of lower and upper bounds on smoothness, which are to some extent surprising: (1) The smoothness of a single block network is lg w + Theta(1) (with high probability), where the additive constant is between -2 and 3. This tight bound improves vastly over the upper bound of O(root lg w) from Herlihy and Tirthapura (2006), and it essentially settles our understanding of the smoothing properties of the block network. Further, this lg w + Theta (1) bound cannot be extended to universal smoothing, a property stronger than smoothing that we introduce. (2) Most significantly, the smoothness of the cascade of two block networks is no more than 17 (with high probability); this is the first known randomized network with so small depth (2 lgw) and so good (constant) smoothness. The proof introduces some combinatorial and probabilistic structures and techniques which may be further applicable; the result demonstrates the full power of randomization in smoothing networks. (3) There is no randomized 1-smoothing network of width w and depth d that achieves 1-smoothnesswith probability better than d w -1. In view of the deterministic, (lg w)-depth, 1-smoothing network in Klugerman and Plaxton (Proceedings of the 24th Annual ACM Symposium on Theory of Computing, 1992), this result implies the first separation between deterministic and randomized smoothing networks. These results demonstrate an unexpected limitation on the power of randomization for smoothing networks: although it yields constant smoothness using small depth, going down to smoothness of 1 requires linear depth.