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The impact of randomization in smoothing networks

Published: 18 August 2008 Publication History

Abstract

We revisit smoothing networks which are made up of balancers and wires. Tokens arrive arbitrarily on w input wires and propagate asynchronously through the network; each token gets service 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 assume that balancers are oriented independently and uniformly at random. We present a collection of lower and upper bounds on smoothness, which are to some extent surprising:-The smoothness of a single block network is log log w + Θ(1) (with high probability), where the additive constant is between -2 and 4. This tight bound improves vastly over the upper bound of O(√log w) from Herlihy and Tirthapura, and it significantly improves our understanding of the smoothing properties of the block network. -Most significantly, the smoothness of the cascade of two block networks is no more than 16 (with high probability); this is the first known randomized network with so small depth (2 log w) and so good smoothness. The proof introduces some novel combinatorial and probabilistic structures and techniques which may be further applicable. This result demonstrates the full power of randomization in smoothing networks. -There is no randomized 1-smoothing network of width w and depth d that achieves 1-smoothness with probability better than d/w-1. In view of the deterministic 1-smoothing network from Klugerman and Plaxton, this result implies the first separation between deterministic and randomized smoothing networks, which demonstrates an unexpected limitation of randomization: it can get to constant smoothness very easily, but after that, the progress to 1-smoothing is very limited.

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Cited By

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  • (2010)An efficient counting networkTheoretical Computer Science10.1016/j.tcs.2010.04.023411:34-36(3001-3030)Online publication date: 1-Jul-2010
  • (2009)A randomized, o(log w)-depth 2 smoothing networkProceedings of the twenty-first annual symposium on Parallelism in algorithms and architectures10.1145/1583991.1584043(178-187)Online publication date: 11-Aug-2009
  • (2009)Near-perfect load balancing by randomized roundingProceedings of the forty-first annual ACM symposium on Theory of computing10.1145/1536414.1536433(121-130)Online publication date: 31-May-2009
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cover image ACM Conferences
PODC '08: Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
August 2008
474 pages
ISBN:9781595939890
DOI:10.1145/1400751
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 18 August 2008

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  1. randomization
  2. smoothing networks

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Cited By

View all
  • (2010)An efficient counting networkTheoretical Computer Science10.1016/j.tcs.2010.04.023411:34-36(3001-3030)Online publication date: 1-Jul-2010
  • (2009)A randomized, o(log w)-depth 2 smoothing networkProceedings of the twenty-first annual symposium on Parallelism in algorithms and architectures10.1145/1583991.1584043(178-187)Online publication date: 11-Aug-2009
  • (2009)Near-perfect load balancing by randomized roundingProceedings of the forty-first annual ACM symposium on Theory of computing10.1145/1536414.1536433(121-130)Online publication date: 31-May-2009
  • (2009)Smoothed Analysis of Balancing NetworksProceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II10.1007/978-3-642-02930-1_39(472-483)Online publication date: 3-Jul-2009

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