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Smoothed analysis of balancing networks
Author(s) -
Friedrich Tobias,
Sauerwald Thomas,
Vilenchik Dan
Publication year - 2011
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20341
Subject(s) - security token , combinatorics , upper and lower bounds , computer science , binary logarithm , mathematics , discrete mathematics , mathematical analysis , computer security
In a balancing network each processor has an initial collection of unit‐size jobs (tokens) and in each round, pairs of processors connected by balancers split their load as evenly as possible. An excess token (if any) is placed according to some predefined rule. As it turns out, this rule crucially affects the performance of the network. In this work we propose a model that studies this effect. We suggest a model bridging the uniformly‐random assignment rule, and the arbitrary one (in the spirit of smoothed‐analysis). We start with an arbitrary assignment of balancer directions and then flip each assignment with probability α independently. For a large class of balancing networks our result implies that after \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs}\pagestyle{empty}\begin{document} $\mathcal{O}(\log n)$ \end{document} rounds the discrepancy is \documentclass{article} \usepackage{amsmath,amsfonts,mathrsfs}\pagestyle{empty}\begin{document} $\mathcal{O}( (1/2-\alpha) \log n + \log \log n)$ \end{document} with high probability. This matches and generalizes known upper bounds for α = 0 and α = 1/2. We also show that a natural network matches the upper bound for any α. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 115–138, 2011