A combinatorial treatment of balancing networks

Balancing networks, originally introduced by Aspnes et al. (Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348-358, May 1991), represent a new class of distributed, low-contention data structures suitable for solving many fundamental multi-processor coordination problems that can be expressed as balancing problems. In this work, we present a mathematical study of the combinatorial structure of balancing networks, and a variety of its applications. Our study identifies important combinatorial transfer parameters of balancing networks. In turn, necessary and sufficient combinatorial conditions are established, expressed in terms of transfer parameters, which precisely characterize many important and well studied classes of balancing networks such as counting networks and smoothing networks. We propose these combinatorial conditions to be "balancing analogs" of the well known Zero-One principle holding for sorting networks. Within the combinatorial framework we develop, our first application is in deriving combinatorial conditions, involving the transfer parameters, which precisely delimit the boundary between counting networks and sorting networks.