Abstract

Balancing networks, originally introduced by Aspnes et al. (Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348-358, May 1991), represented a new class of distributed, low-contention data structures suitable for solving many fundamental multi-processor coordination problems that can be expressed asbalancing 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 parametersof 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 ascounting networksandsmoothing networks. We propose these combinatorial conditions to be ``balancing analogs'' of the well knownZero-One principleholding forsorting 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.

We next turn to use the necessity of the shown combinatorial conditions in deriving width inconstructibility results and lower bounds on depth-like measures for several classes of balancing networks; these results significantly improve upon previous ones shown by Aharonson and Attiya (

Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 104-113, January 1992), and Moran and Taubenfeld (Proceedings of the 12th Annual ACM Symposium on Principles of Distributed Computing, pp. 251-259, August 1993) in terms of strength, generality, and proof simplicity.The sufficiency of the shown combinatorial conditions is employed in designing the first formal algorithms for mathematically verifying that a given network belongs to each of a collection of classes. These algorithms are simple, modular and easy to implement, consisting merely of multiplying matrices and evaluating matricial functions.

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Preliminary versionA preliminary version of these results was presented in: Costas Busch and Marios Mavronicolas. A combinatorial treatment of balancing networks. In Proceedings of the Thirteenth Annual ACM Symposium on Principles of Distributed Computing, pages 206-215, Los Angeles, California, 14-17 August 1994.

Categories and Subject Descriptors: C.2.1 [Computer-Communication Networks]: Network Architecture and Design --distributed networks,network topology; C.2.4 [Computer-Communication Networks]: Distributed Systems --distributed applications; F.1.2 [Computation by Abstract Devices]: Modes of Computation --parallelism and concurrency; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems --sorting and searching; G.2.1 [Discrete Mathematics]: Combinatorics --combinatorial algorithms,counting problems; G.2.2 [Discrete Mathematics]: Graph Theory --graph algorithms,networks problems

General Terms: Theory, Verification, Algorithms

Additional Key Words and Phrases: Balancing networks, block-input networks, block-output networks, combinatorial characterizations, counting networks, impossibility results, incidence matrices, smoothing networks, transfer parameters

Selected papers that cite this one

- Costas Busch and Marios Mavronicolas. Impossibility results for weak threshold networks. Information Processing Letters, 63(2):85-90, 28 July 1997.

Selected references

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