Abstract
Finding the singularity of a matrix is a basic problem in linear algebra. Chu and Schnitger [SC95] first considered this problem in the communication complexity model, in which Alice holds the first half of the matrix and Bob holds the other half. They proved that the deterministic communication complexity is Omega(n^2 log p) for an n by n matrix over the finite field F_p. Then, Clarkson and Woodruff [CW09] introduced the singularity problem to the streaming model. They proposed a randomized one pass streaming algorithm that uses O(k^2 log n) space to decide if the rank of a matrix is k, and proved an Omega(k^2) lower bound for randomized oneway protocols in the communication complexity model.
We prove that the randomized/quantum communication complexity of the singularity problem over F_p is Omega(n^2 log p), which implies the same space lower bound for randomized streaming algorithms, even for a constant number of passes. The proof uses the framework by Lee and Shraibman [LS09], but we choose Fourier coefficients as the witness for the dual approximate norm of the communication matrix. Moreover, we use Fourier analysis to show the same randomized/quantum lower bound when deciding if the determinant of a nonsingular matrix is a or b for nonzero a and b.
BibTeX  Entry
@InProceedings{sun_et_al:LIPIcs:2012:3438,
author = {Xiaoming Sun and Chengu Wang},
title = {{Randomized Communication Complexity for Linear Algebra Problems over Finite Fields}},
booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
pages = {477488},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897354},
ISSN = {18688969},
year = {2012},
volume = {14},
editor = {Christoph D{\"u}rr and Thomas Wilke},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2012/3438},
URN = {urn:nbn:de:0030drops34385},
doi = {10.4230/LIPIcs.STACS.2012.477},
annote = {Keywords: communication complexity, streaming, matrix, singularity, determinant}
}
Keywords: 

communication complexity, streaming, matrix, singularity, determinant 
Collection: 

29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012) 
Issue Date: 

2012 
Date of publication: 

24.02.2012 