scholarly journals A quadratic lower bound for homogeneous algebraic branching programs

2019 ◽  
Vol 28 (3) ◽  
pp. 409-435
Author(s):  
Mrinal Kumar
Keyword(s):  
Lower Bound ◽  
Branching Programs ◽  
Algebraic Branching Programs

On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models

2020 ◽  
Vol 177 (1) ◽  
pp. 69-93
Author(s):  
Purnata Ghosal ◽  
B.V. Raghavendra Rao
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Partial Derivative ◽  
Computable Function ◽  
Arithmetic Circuits ◽  
Multilinear Polynomial ◽  
Algebraic Models ◽  
Branching Programs ◽  
Algebraic Branching Programs ◽  
Derivative Matrix

We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. We consider the following special classes of multilinear algebraic branching programs: 1) Read Once Oblivious Branching Programs (ROABPs), 2) Strict interval branching programs, 3) Sum of read once formulas with restricted ordering. We obtain parameterized lower bounds (i.e., nΩ(t(k)) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size g(k)nO(1) for some computable function g. Further, we obtain a parameterized separation between ROABPs and read-2 ABPs. This is obtained by constructing a degree k polynomial that can be computed by a read-2 ABP of small size such that the rank of the partial derivative matrix under any partition of the variables is large.

scholarly journals A lower bound for read-once-only branching programs

1987 ◽  
Vol 35 (2) ◽  
pp. 153-162 ◽  
Author(s):  
László Babai ◽  
Péter Hajnal ◽  
Endre Szemerédi ◽  
György Turán
Keyword(s):  
Lower Bound ◽  
Branching Programs

scholarly journals On Ajtai’s Lower Bound Technique for R-way Branching Programs and the Hamming Distance Problem

2000 ◽  
Vol 7 (11) ◽  
Author(s):  
Jakob Pagter
Keyword(s):  
Lower Bound ◽  
Hamming Distance ◽  
Linear Time ◽  
Program Model ◽  
Branching Programs ◽  
Original Proof ◽  
Trade Off ◽  
Time Space ◽  
Distance Problem ◽  
With Memory

In this report we study the proof employed by Miklos Ajtai<br />[Determinism versus Non-Determinism for Linear Time RAMs<br />with Memory Restrictions, 31st Symposium on Theory of <br />Computation (STOC), 1999] when proving a non-trivial lower bound<br />in a general model of computation for the Hamming Distance<br />problem: given n elements: decide whether any two of them have<br />"small" Hamming distance. Specifically, Ajtai was able to show<br />that any R-way branching program deciding this problem using<br />time O(n) must use space Omega(n lg n).<br />We generalize Ajtai's original proof allowing us to prove a<br />time-space trade-off for deciding the Hamming Distance problem<br /> in the R-way branching program model for time between n<br />and alpha n lg n / lg lg n, for some suitable 0 < alpha < 1. In particular we prove<br />that if space is O(n^(1−epsilon)), then time is Omega(n lg n / lg lg n).

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