Linear sketching over F<inf>2</inf>
Name
LIPIcs-CCC-2018-8.pdf
Description
Published version
Size
796.91 KB
Format
Adobe PDF
Checksum (MD5)
03f7704611178c1298b59e9d43bb2a5c
Author(s)
Kannan, Sampath
Mossel, Elchanan
Sanyal, Swagato
Mossel, Elchanan
Date Issued
June 2018
Citation
Kannan, Sampath, Mossel, Elchanan, Sanyal, Swagato and Mossel, Elchanan. 2018. "Linear sketching over F2."
Version
Final published version
Abstract
© Sampath Kannan, Elchanan Mossel, Swagato Sanyal, and Grigory Yaroslavtsev; licensed under Creative Commons License CC-BY 33rd Computational Complexity Conference (CCC 2018). We initiate a systematic study of linear sketching over F2. For a given Boolean function treated as f: F2 → F 2 a randomized F2-sketch is a distribution M over d × n matrices with elements over F2 such that Mx suffices for computing f(x) with high probability. Such sketches for d << n can be used to design small-space distributed and streaming algorithms. Motivated by these applications we study a connection between F2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F2-sketching is optimal (up to constant factors) for uniformly distributed inputs. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F2 can be constructed as F2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length Õ(n) constructed through uniformly random updates.
MIT Department
Massachusetts Institute of Technology
Terms of Use
Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.4230/LIPIcs.CCC.2018.8
