Lower Bounds for Threshold and Symmetric Functions in Parallel Computation

1992 ◽  
Vol 21 (2) ◽  
pp. 329-338 ◽  
Author(s):  
Yossi Azar
Keyword(s):  
Parallel Computation ◽  
Lower Bounds ◽  
Symmetric Functions

scholarly journals Lower Bounds for Dynamic Algebraic Problems

1998 ◽  
Vol 5 (11) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Johan P. Hansen ◽  
Peter Bro Miltersen
Keyword(s):  
Fourier Transform ◽  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Symmetric Functions ◽  
Matrix Multiplication ◽  
Worst Case ◽  
Dynamic Matrix ◽  
Wide Range ◽  
On Line

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form "change input xi to value v" or "what is the value of output yi?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent<br />algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal  Omega(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an <br />Omega(sqrt(n)) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and<br />Tate's O(sqrt(n log n)) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Omega(sqrt(n)) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting<br />of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an ((log n)2= log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields.

New lower bounds for parallel computation

1989 ◽  
Vol 36 (3) ◽  
pp. 671-680 ◽  
Author(s):  
Ming Li ◽  
Yaacov Yesha
Keyword(s):  
Parallel Computation ◽  
Lower Bounds

Upper and Lower Bounds for The Area of a Triangle for Whose Sides Two Symmetric Functions are Known

1957 ◽  
Vol 9 ◽  
pp. 227-231 ◽  
Author(s):  
Robert Frucht
Keyword(s):  
No Value ◽  
Lower Bounds ◽  
Lower Estimate ◽  
Symmetric Functions ◽  
Upper And Lower Bounds ◽  
Image Position

Improving on inequalities given by Gerretsen (2), Beatty (1) has proved that for the area Δ of any plane triangle with sides a, b, c the following inequalities hold:(1.1)where(1.2);the signs of equality in (1.1) only apply when the triangle is equilateral. Beatty has also remarked that the second inequality in (1.1) is of no value in case 5H ≥ 3K, since then the lower estimate which it gives for Δ2 is not even positive.

scholarly journals Shuffles and Circuits (On Lower Bounds for Modern Parallel Computation)

2018 ◽  
Vol 65 (6) ◽  
pp. 1-24 ◽  
Author(s):  
Tim Roughgarden ◽  
Sergei Vassilvitskii ◽  
Joshua R. Wang
Keyword(s):  
Parallel Computation ◽  
Lower Bounds
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