1. Divide-and-Conquer Design
Ladner-Fischer construction
T(n) = T(n/2) + 1
= log2n
C(n) = 2C(n/2) + n/2
= (n/2) log2n
Simple Ladner-Fisher
Parallel prefix network
(its delay is optimal,
but has fan-out issues
if implemented directly)
Prefix sum network built of two n/2-input networks and n/2 adders.
Fall 2008
Parallel Processing, Extreme Models

2. 8.4 Parallel Prefix Networks
T(n) = T(n/2) + 2
= 2 log2n – 1
C(n) = C(n/2) + n – 1
= 2n – 2 – log2n
This is the Brent-Kung
Parallel prefix network
(its delay is actually
2 log2n – 2)
Fig Prefix sum network built of one n/2-input network and n – 1 adders.
Fall 2008
Parallel Processing, Extreme Models

3. Example of Brent-Kung Parallel Prefix Network
Originally developed
by Brent and Kung as
part of a VLSI-friendly
carry lookahead adder
One level of latency
T(n) = 2 log2n – 2
C(n) = 2n – 2 – log2n
Brent–Kung parallel prefix graph for n = 16.
Fall 2008
Parallel Processing, Extreme Models

4. Example of Kogge-Stone Parallel Prefix Network
T(n) = log2n
C(n) = (n – 1) + (n – 2)
+ (n – 4) n/2
= n log2n – n + 1
Optimal in delay,
but too complex
in number of cells
and wiring pattern
Kogge-Stone parallel prefix graph for n = 16.
Fall 2008
Parallel Processing, Extreme Models

5. Comparison and Hybrid Parallel Prefix Networks
Brent/Kung
6 levels
26 cells
Kogge/Stone
4 levels
49 cells
Fig A hybrid Brent–Kung / Kogge–Stone parallel prefix graph for n = 16.
Han/Carlson
5 levels
32 cells
Fall 2008
Parallel Processing, Extreme Models

6. Another Divide-and-Conquer Algorithm
T(p) = T(p/2) + 1
T(p) = log2 p
Strictly optimal algorithm, but requires commutativity
Each vertical line represents a location in shared memory
Another divide-and-conquer scheme for parallel prefix computation.
Fall 2010
Parallel Processing, Shared-Memory Parallelism

7. کاربردهایی از شبکه جمع پیشوندی
شماره بیتهای یک
کشف بیت یک دارای اولویت

8. n-th Root of Unity
Xn = 1
wnk+n/2 = -wnk
wndkd = wnk

9. DFT, DFT-1
O(n2)-time naïve algorithm using Horner-evaluation

10. Application of DFT to Smoothing or Filtering
Fall 2008
Parallel Processing, Extreme Models

11. Application of DFT to Spectral Analysis
Fall 2008
Parallel Processing, Extreme Models

12. Multiplying Two Polynomial

13. Multiplying Two Polynomial

15. Fast Fourier Transform (FFT)

16. Parallel Processing, Extreme Models
Fast Fourier Transform (FFT)
yi = ui + wni vi (0  i < n/2)
yi+n/2 = ui + wni+n/2 vi = ui - wni vi
T(n) = 2T(n/2) + n = n log2n sequentially
T(n) = T(n/2) + 1 = log2n in parallel
Fall 2008
Parallel Processing, Extreme Models

20. Computation Scheme for 16-Point FFT
Fall 2008
Parallel Processing, Extreme Models

21. Parallel Architectures for FFT
Butterfly network for an 8-point FFT.
Parallel Processing, Extreme Models