1. Block p and g Generators

2. Carry Determination as Prefix Computations Two Contiguous (or Overlapping) Blocks (g’, p’) and (g’’, p’’) Merged Block (g, p) g = g’’+ g’p’’ p = p’p’’ Large Group Generates Carry if: 1)left group generates carry 2)right group generates and left group propagates

3. Carry Operator, ¢ Define Operator Over (g, p) Pairs (g, p) = (g’, p’) ¢ (g’’, p’’) g = g’’+ g’p’ p = p’p’’ ¢ is Associative (g’, p’) ¢ (g’’, p’’) ¢ (g’’’, p’’’) =[(g’, p’) ¢ (g’’, p’’) ] ¢ (g’’’, p’’’) = (g’, p’) ¢ [(g’’, p’’) ¢ (g’’’, p’’’)]

4. Carry Operator, ¢ (cont) ¢ is NOT Commutative (g’, p’) ¢ (g’’, p’’)  (g’’, p’’)¢ (g’, p’) This is Easy to See Because: g = g’’+ g’p’  g’+ g’’p’’

5. Prefix Adders

6. Carry Determination Assume Adder with NO c IN c i+1 = g [0,i] Carry Enters i+1 Block iff Generated in Block [0,i] Assume Adder with c IN = 1 Viewed as Generated Carry from Stage -1 p -1 = 0, g -1 = c IN Compute g [-1,i] For All i Formulate Carry Determination as:

7. Prefix Computation

8. Prefix Sums Analogy Designs for Prefix Sums Can be Converted to Carry Computation Replace Adder with ¢ Operator Addition IS Commutative, Order Doesn’t Matter Can Group (g, p) In Anyway to Combine Into Block Signals (as long as order is preserved) (g, p) Allow for Overlapping Groups, Prefix Sums Does Not (sum would contain some values added two or times)

9. Prefix Sum Network (adder levels) (# of adders)

10. Another Way for Prefix Sums Compute the Following First: x 0 +x 1 x 2 +x 3 x 4 +x 5...x k-2 +x k-1 Yields the Partial Sums, s 1, s 3, s 5,..., s k-1 Next, Even Indexed Sums Computed As: s 2j = s 2j-1 + x 2j

11. Alternative Prefix Sum Network

12. Comparison of Prefix Sum Networks First Design Faster: lg 2 (k) versus 2lg 2 (k)-2(levels) First Design has High Fan-out Requirements First Design Requires More Cells (k/2)lg 2 k versus 2k-2-lg 2 k Second Design is Brent-Kung Parallel Prefix Graph First Design is Kogge-Stone Parallel Prefix Graph (fan-out can be avoided by distributing computations)

13. Brent-Kung Network independent, so single delay

14. Kogge-Stone Network

15. Area/Levels of Prefix Networks

16. Hybrid Parallel Prefix Network