1. Carry Look Ahead (CLA)

2. Full Adder Bi Ai Cin Cout Si
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Full Adder Bi Ai
Cin
Cout Si

3. Carry Ripple Adder A0 B0 S0 A1 B1 S1 A2 B2 S2 A3 B3 S3 C0=0 C1 C2 C3
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Carry Ripple Adder A0 B0 S0 A1 B1 S1 A2 B2 S2 A3 B3 S3
C0=0 C1 C2 C3 C4

4. © Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

5. Reduction Changing the goal to: Calculate the Carry vector
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Reduction
Changing the goal to: Calculate the Carry vector
We can calculate S out of A,B,C in O(1)
We’d like to improve the complexity of calculating C (less than linear)

6. Three Cases 1 Generate: Gi ? 1 Propagate: Pi ? Kill  Gi  Pi ?
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Three Cases 1 ?
Generate: Gi 1 ?
Propagate: Pi ?
Kill  Gi  Pi

7. חישובCout בהינתן Cin ותכונה
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
חישובCout בהינתן Cin ותכונה
תכונה G P
Cout
Generate 1
Propogate
Cin
Kill Gi Pi
Ci+1= Gi V (Pi Λ Ci) Ci
Ci+1

8. G/P gate Gi Pi Gi 1 Pi 1 Pi 1 Ai Bi Gi = Ai Λ Bi Pi = Ai  Bi
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
G/P gate Gi 1 Pi 1 Pi 1 Ai Bi Gi Pi
Gi = Ai Λ Bi
Pi = Ai  Bi

9. Calculate using 2 G/P pairsCi Gi Pi
Gi+1
Pi+1
Ci+1
Ci+2
Ci+2=1  Gi+1 V (Gi Λ Pi+1)
Ci+2=Ci  Pi Λ Pi+1
Ci+2=0  otherwise Gi Pi G’ P’
Gi+1
Pi+1
G’ = Gi+1 V (Gi Λ Pi+1)
P’ = Pi Λ Pi+1
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

10. חישוב וקטורי G ו-P: O(log(n))A0 B0 A1 B1 A2 B2 A3 B3 A4 B4 A5 B5 A6 B6 A7 B7
G0-1P0-1
G2-3P2-3
G4-5P4-5
G6-7P6-7
G0-3P0-3
G4-7P4-7 G0 P0 G1 P1 G2 P2 G3 P3 G4 P4 G5 P5 G6 P6 G7 P7
חישוב וקטורי G ו-P: O(log(n))
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

11. חישוב של Cj+1 בהינתן Ci ובלוק Gi-jPi-j
Cj+1 = Gi-j V (Pi-j Λ Ci)
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

12. תלות של Cj+1 ב-Ci ,Gi-j ו-Pi-j
Cj+1 = Cj+1(Ci , Gi-j , Pi-j) = Gi-j V (Pi-j Λ Ci) C7 C5 C3 C1 C6 C4 C2
C0=0 C6 C2
G6-7P6-7
G4-5P4-5
G2-3P2-3
G0-1P0-1 C4
C0=0 C8
C0=0 C4
G4-7P4-7
G0-3P0-3
חישוב ה-carries בהינתן וקטורי G ו-P: O(log(n))
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

13. log(n) +log(n) = 2log(n) = O(log(n))
C0=0 A0 B0 A1 B1 A2 B2 A3 B3 A4 B4 A5 B5 A6 B6 A7 B7 G0 P0 G1 P1 G2 P2 G3 P3 C1 C2 C3 C4 G4 P4 G5 P5 G6 P6 G7 P7 C5 C6 C7 C8
G0-1P0-1
G2-3P2-3
G4-5P4-5
G6-7P6-7
G0-3P0-3
G4-7P4-7
log(n) +log(n) = 2log(n) = O(log(n))
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

14. אין מסלול מ-Ci ל-G או P Gj Pj Gi Pi Gj Pj Gi Pi G0-1P0-1 Gi-j Pi-j
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
אין מסלול מ-Ci ל-G או P Gj Pj Gi Pi Gj Pj Gi Pi
G0-1P0-1
Gi-j
Pi-j
G0-1P0-1
Cj+1 Ci
Gi-j
Pi-j Ci
Cj+1

15. Timeline of Calculations for n=8
C0 is 0 from the beginning.
1) GiPi for each 0 ≤ i < 8
2) C1 (by C0 and G0P0)
G0-1P0-1, G2-3P2-3,
G4-5P4-5, G6-7P6-7
3) C2 (by C0 and G0-1P0-1)
G0-3P0-3, G4-7P4-7
// Now all G...P... are known Ai Bi Gi Pi
4) C3 (by C2 and G2P2)
C4 (by C0 and G0-3P0-3)
5) C5 (by C4 and G4P4)
C6 (by C4 and G4-5P4-5)
6) C7 (by C6 and G6P6)
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

16. Final stage A0 B0 S0 C0 A1 B1 S1 C1 A2 B2 S2 C2
An-1
Bn-1
Sn-1
Cn-1
... Cn
Overall complexity: O(log(n))+1=O(log(n)) = 2⋅log2n + 1
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology

17. No! 2∙log2n + 1 < n n ≥ 6 Example 2 from .pdf Ripple-Carry n
Adder
Calculation Time
Ripple-Carry n
Carry-Look-Ahead
2∙log2n + 1
CLA is better when:
2∙log2n + 1 < n
n ≥ 6
© Dima Elenbogen 2010, Technion – Israel Institute of Technology

18. למעשה אותה פונקציה בוליאנית
שאלה
© Yohai Devir 2007, Dima Elenbogen 2011, Technion – Israel Institute of Technology
רשמו את כל הפונקציות הבוליאניות מהצורה :{0,1}←{0,1}* המתארות את היחידות של מחבר CLA.
למעשה אותה פונקציה AND
למעשה אותה פונקציה AND
למעשה אותה פונקציה בוליאנית
Si = Ai  Bi  Ci

19. Example 3 from .pdf G* = Gi+2 + Gi+1Pi+2 + GiPi+1Pi+2
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Example 3 from .pdf Ci Gi Pi
Gi+1
Pi+1
Ci+1
Gi+2
Pi+2
Ci+3
Ci+2
G* = Gi+2 + Gi+1Pi+2 + GiPi+1Pi+2
P* = Pi  Pi+1  Pi+2

20. Comparator A==B, A>B
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Comparator A==B, A>B
Propogate
Generate A0 B0 A1 B1 A2 B2 A3 B3
EQ4
EQ1
EQ2
EQ3
EQ0=1
GR4
GR1
GR2
GR3
GR0=0

21. Comparator A==B, A>B
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology
Comparator A==B, A>B A0 B0 A1 B1
EQ0-1
GR0-1
EQ0
GR0
GR1
EQ1
EQ0-1  A0A1==B0B1
EQ0-1 = EQ1 Λ EQ0
GR0-1  A0A1 > B0B1
GR0-1 = GR1 V (GR0 Λ EQ1)

22. log(n) + 1 = O(log(n)) A0 B0 A1 B1 A2 B2 A3 B3 A4 B4 A5 B5 A6 B6 A7 B7
© Yohai Devir 2007, Dima Elenbogen 2010, Technion – Israel Institute of Technology A0 B0 A1 B1 A2 B2 A3 B3 A4 B4 A5 B5 A6 B6 A7 B7
log(n) + 1 = O(log(n))