A non-linear time lower bound for boolean branching programs (1999)
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| Venue: | In Proc. of 40th FOCS |
| Citations: | 53 - 0 self |
BibTeX
@INPROCEEDINGS{Ajtai99anon-linear,
author = {Miklós Ajtai},
title = {A non-linear time lower bound for boolean branching programs},
booktitle = {In Proc. of 40th FOCS},
year = {1999},
pages = {60--70},
publisher = {IEEE}
}
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Abstract
Abstract: We give an exponential lower bound for the size of any linear-time Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδ|logδ | −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.
