scholarly journals Beating the Direct Sum Theorem in Communication Complexity with Implications for Sketching

2013 ◽  
Author(s):  
Marco Molinaro ◽  
David P. Woodruff ◽  
Grigory Yaroslavtsev
Keyword(s):  
Communication Complexity ◽  
Direct Sum ◽  
Sum Theorem

Super-logarithmic depth lower bounds via the direct sum in communication complexity

1995 ◽  
Vol 5 (3-4) ◽  
pp. 191-204 ◽  
Author(s):  
Mauricio Karchmer ◽  
Ran Raz ◽  
Avi Wigderson
Keyword(s):  
Lower Bounds ◽  
Communication Complexity ◽  
Direct Sum

On the Decomposition of Continuous Modules

1982 ◽  
Vol 25 (3) ◽  
pp. 296-301 ◽  
Author(s):  
Bruno J. Müller ◽  
S. Tariq Rizvi
Keyword(s):  
Decomposition Theorem ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Sum Theorem ◽  
Continuous Module

AbstractWe prove two theorems on continuous modules:Decomposition Theorem. A continuous moduleMhas a decomposition,M=M1⊕M2, such thatM1is essential over a direct sumof indecomposable summandsAiofM, andM2has no uniform submodules; and these data are uniquely determined byMup to isomorphism.Direct Sum Theorem. A finite direct sumof indecomposable modulesAiis continuous if and only if eachAiis continuous andAj-injective for allj≠ i.

scholarly journals The quantum query complexity of certification

2010 ◽  
Vol 10 (3&4) ◽  
pp. 181-189
Author(s):  
A. Ambainis ◽  
A.M. Childs ◽  
F. Le Gall ◽  
S. Tani
Keyword(s):  
Lower Bound ◽  
Query Complexity ◽  
Direct Sum ◽  
Quantum Query Complexity ◽  
Sum Theorem ◽  
Adversary Method ◽  
Quantum Query

We study the quantum query complexity of finding a certificate for a d-regular, k-level balanced \nand formula. We show that the query complexity is $\tilde\Theta(d^{(k+1)/2})$ for 0-certificates, and $\tilde\Theta(d^{k/2})$ for 1-certificates. In particular, this shows that the zero-error quantum query complexity of evaluating such formulas is $\tilde O(d^{(k+1)/2})$. Our lower bound relies on the fact that the quantum adversary method obeys a direct sum theorem.

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