On the Communication Complexity of Read-Once AC^0 Formulae
Name
Jayram-2009-On the communication complexity of read-once ACO formulae.pdf
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419.61 KB
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Author(s)
Kopparty, Swastik
Raghavendra, Prasad
Jayram, T. S
Alternative Title
On the communication complexity of read-once ACO formulae
Date Issued
September 2009
Journal
24th Annual IEEE Conference on Computational Complexity, 2009. CCC '09.
Publisher
Institute of Electrical and Electronics Engineers
Citation
Jayram, T.S., S. Kopparty, and P. Raghavendra. “On the Communication Complexity of Read-Once AC^0 Formulae.” Computational Complexity, 2009. CCC '09. 24th Annual IEEE Conference on. 2009. 329-340. © 2009 IEEE
Version
Final published version
Abstract
We study the 2-party randomized communication complexity of read-once AC[superscript 0] formulae. For balanced AND-OR trees T with n inputs and depth d, we show that the communication complexity of the function f[superscript T](x, y) = T(x omicron y) is Omega(n/4[superscript d]) where (x omicron y)[subscript i] is defined so that the resulting tree also has alternating levels of AND and OR gates. For each bit of x, y, the operation omicron is either AND or OR depending on the gate in T to which it is an input. Using this, we show that for general AND-OR trees T with n inputs and depth d, the communication complexity of f[superscript T](x, y) is n/2[superscript Omega(d log d)]. These results generalize classical results on the communication complexity of set-disjointness (where T is an OR -gate) and recent results on the communication complexity of the TRIBES functions (where T is a depth-2 read-once formula). Our techniques build on and extend the information complexity methodology for proving lower bounds on randomized communication complexity. Our analysis for trees of depth d proceeds in two steps: (1) reduction to measuring the information complexity of binary depth-d trees, and (2) proving lower bounds on the information complexity of binary trees. In order to execute this program, we carefully construct input distributions under which both these steps can be carried out simultaneously. We believe the tools we develop will prove useful in further studies of information complexity in particular, and communication complexity in general.
Subjects
Lower bounds
Information complexity
Communication complexity
AND-OR trees
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://dx.doi.org/10.1109/CCC.2009.39
