This document discusses parallel prefix adders. It provides background on parallel prefix operations and defines binary addition as a parallel prefix problem. The key steps of carry lookahead adders are described, including precomputing propagate and generate values, calculating carries through a carry generation block, and combining carries and propagates to generate the sum. Several parallel prefix adder architectures are introduced, including the Sklansky conditional adder, Kogge-Stone adder, and Ladner-Fischer adder, which aim to optimize parameters like depth, node count, and fan-out.

1. Parallel prefix
adders
Kostas Vitoroulis, 2006.
Presented to Dr. A. J. Al-Khalili.
Concordia University.

2. Overview of presentation
 Parallel prefix operations
 Binary addition as a parallel prefix
operation
 Prefix graphs
 Adder topologies
 Summary

3. Parallel Prefix Operation
Terminology background:
 Prefix: The outcome of the operation depends on the initial inputs.
 Parallel: Involves the execution of an operation in parallel. This is
done by segmentation into smaller pieces that are computed in
parallel.
 Operation: Any arbitrary primitive operator “ ° ” that is associative
is parallelizable
 it is fast because the processing is accomplished in a parallel fashion.

4. Example: Associative operations are parallelizable
Consider the logical OR operation: a + b
The operation is associative:
a + b + c + d = ((( a + b ) + c) + d ) = (( a + b ) + ( c + d))
Serial implementation: Parallel implementation:
a+b a a+b
b
(a+b)+(c+d)
(a+b)+c
c
d c+d
((a+b)+c)+d

5. Mathematical Formulation: Prefix Sum
 Operator: “ ° ”  this is the unary operator
known as “scan” or “prefix
sum”
 Input is a vector:
A = AnAn-1 … A1
 Output is another vector:
B = BnBn-1 … B1
where
B1 = A1  Bn represents the
B2 = A1 ° A2 operator being applied to
… all terms of the vector.
Bn = A1 ° A2 … ° An

6. Example of prefix sum
Consider the vector: A = AnAn-1 … A1 where element Ai is an integer
The “*” unary operator, defined as:
*A = B
With
B = BnBn-1 … B1
B 1 = A1
B2 = A1 * A2
B 3 = A 1 * A 1 * A3
…
and ‘ * ’ here is the integer addition operation.

7. Example of prefix sum
Calculation of *A, where A = 6 5 4 3 2 1 yields:
B = *A = 21 15 10 6 3 1
Because the summation is associative the calculation can be done in parallel in the
following manner:
Parallel implementation versus Serial implementation
6 5 4 3 2 1 6 5 4 3 2 1
+ + + +
+ +
B = (A + A +3A3
B6 =3B61+…A1 + A2) =
A 2 = A11 = 12
+A +
1
= (A6 + A+) +
+ + =6 5
((A4+A3) +(A2 +A1))
+
+= 21
B6 B5 B4 B3 B2 B1 B6 B5 B4 B3 B2 B1

8. Binary Addition This is the pen and paper addition of
two 4-bit binary numbers x and y.
c represents the generated carries.
s represents the produced sum bits.
c3 c2 c1 c0
x3 x2 x1 x0 A stage of the addition is the set of
+ x and y bits being used to produce
y3 y2 y1 y0 the appropriate sum and carry bits.
s4 s3 s2 s1 s0 For example the highlighted bits x2,
y2 constitute stage 2 which
generates carry c2 and sum s2 .
Each stage i adds bits ai, bi, ci-1 and produces bits si, ci
The following hold:
ai bi ci Comment: Formal definition:
0 0 0 The stage “kills” an incoming carry. “Kill” bit: ki = xi + yi
0 1 ci-1 The stage “propagates” an incoming carry “Propagate” bit:
pi = xi ⊕ yi
1 0 ci-1 The stage “propagates” an incoming carry
1 1 1 The stage “generates” a carry out “Generate” bit: g i = xi • yi

9. Binary Addition
ai bi ci Comment: Formal definition:
0 0 0 The stage “kills” an incoming carry. “Kill” bit: ki = xi + yi
0 1 ci-1 The stage “propagates” an incoming carry “Propagate” bit:
pi = xi ⊕ yi
1 0 ci-1 The stage “propagates” an incoming carry
1 1 1 The stage “generates” a carry out “Generate” bit: g i = xi • yi
The carry ci generated by a stage i is given by the equation:
ci = g i + pi ⋅ ci −1 = xi ⋅ yi + ( xi ⊕ yi ) ⋅ ci −1
ci = xi ⋅ yi + ( xi + yi ) ⋅ ci −1 = g i + ai ⋅ ci −1
This equation can be simplified to:
The “ai” term in the equation being the “alive” bit.
The later form of the equation uses an OR gate instead of an XOR which is a more efficient gate when implemented
in CMOS technology. Note that:
ai = ki
Where ki is the “kill” bit defined in the table above.

10. Carry Look Ahead adders
The CLA adder has the following 3-stage structure:
Pre-calculation of pi, gi for each stage
Calculation of carry ci for each stage.
Combine ci and pi of each stage to
generate the sum bits si
Final sum.

11. Carry Look Ahead adders
 The pre-calculation stage is implemented using the
equations for pi, gi shown at a previous slide:
x2y2 x1y1 x0y0
g2 p2 g1 p1 g0 p0
 Alternatively using the “alive” bit:
x2y2 x1y1 x0y0
g2 a2 g1 a1 g0 a0
 Note the symmetry when we use the “propagate” or the “alive” bit… We can use them interchangeably in the equations!

12. Carry Look Ahead adders
 The carry calculation stage is implemented using the
equations produced when unfolding the recursive
equation:
ci = g i + pi ⋅ ci −1 = g i + ai ⋅ ci −1
g2 p2 g1 p1 g0 p0
c0 = g 0
c1 = g1 + p1 ⋅ g 0
c2 = g 2 + p2 ⋅ c1 = g 2 + p2 ⋅ ( g1 + p1 ⋅ g 0 ) Carry generator block
= g 2 + p2 ⋅ g1 + p2 ⋅ p1 ⋅ g 0
etc 
c2 c1 c0

13. Carry Look Ahead adders
 The final sum calculation stage is implemented using the carry and
propagate bits ci,pi:
si = pi ⊕ ci −1 , with pi = xi ⊕ yi
Note :
si = g i + ai ⋅ ci −1 , with ai = xi + yi
c2p3 c1p2 c0p1 cinp0
s3 s2 s1 s0
 If the ‘alive’ bit ai is used the final sum stage becomes more
complex as implied by the equations above.

14. Binary addition as a prefix sum problem.
 We define a new operator: “ ° ”
 Input is a vector of pairs of ‘propagate’ and ‘generate’ bits:
( g n , pn )( g n−1 , pn−1 )  ( g 0 , p0 )
 Output is a new vectornofGn −1 , Pn −1 )  ( G0 , P0 )
( Gn , P )( pairs:
 Each pair of the output vector is calculated by the
following definition: i , Pi ) = ( g i , pi )  (Gi −1 , Pi −1 )
(G
Where :
(G0 , P0 ) = ( g 0 , p0 )
( g x , px )  ( g y , p y ) = ( g x + px ⋅ g y , px ⋅ p y )
with +,⋅ being the OR, AND operations

15. Binary addition as a prefix sum problem.
 Properties of operator “ ° ”:
 Associativity (hence parallelization)
 Easy to prove based on the fact that the logical AND, OR
operations are associative.
 With the definition:
(Gi , Pi ) = ( g i , pi )  (Gi −1 , Pi −1 )
Where (G1 , P ) = ( g1 , p1 )
1
Gi becomes the carry signal at stage i of an adder. Illustration on
next slide.
 The operation is idempotent
( g x , px )  ( g x , px ) = ( g x + px ⋅ g x , px ⋅ px ) = ( g x , px )
 Which implies
(Gi: j , Pi: j ) = (Gi:n , Pi:n )  (Gm: j , Pm: j )
Where i ≥ j and m≥n

16. Binary Addition as a prefix sum problem.
A stage i will generate a carry if
gi=aibi
a3 a2 a1 a0 and propagate a carry if
+ pi=XOR(ai,bi)
b3 b2 b1 b0 Hence for stage i:
ci=gi+pici-1
With : Where :
(Gi , Pi ) = ( g i , pi )  (Gi −1 , Pi −1 ) (G0 , P0 ) = ( g 0 , p0 )
( g x , px )  ( g y , p y ) = ( g x + px ⋅ g y , px ⋅ p y )
… The familiar
We have : carry bit generating
(G1 , P ) = ( g1 , p1 )
1 equations for stage i
(G2 , P2 ) = ( g 2 , p2 )  (G1 , P ) = ( g 2 + p2 ⋅ g1 , p2 ⋅ p1 )
1
in a CLA adder.
(G3 , P3 ) = ( g 3 , p3 )  (G2 , P2 ) = ( g 3 + p3 ⋅ ( g 2 + p2 ⋅ g1 ), p3 ⋅ p2 ⋅ p1 )
= ( g 3 + p3 ⋅ g 2 + p3 ⋅ p2 ⋅ g1 ), p3 ⋅ p2 ⋅ p1 )
etc 

17. Addition as a prefix sum problem.
Conclusion:
The equations of the well known CLA adder can be formulated as a parallel
prefix problem by employing a special operator “ ° ”.
This operator is associative hence it can be implemented in a parallel
fashion.
A Parallel Prefix Adder (PPA) is equivalent to the CLA adder… The two
differ in the way their carry generation block is implemented.
In subsequent slides we will see different topologies for the parallel
generation of carries. Adders that use these topologies are called Parallel
Prefix Adders.

18. Parallel Prefix Adders
 The parallel prefix adder employs the 3-stage structure
of the CLA adder. The improvement is in the carry
generation stage which is the most intensive one:
Pre-calculation of Pi, Gi terms Straight forward as
in the CLA adder
Calculation of the carries. Prefix graphs
can be used to
This part is parallelizable to describe the
reduce time. structure that
performs this
part.
Simple adder to generate the sum Straight forward as
in the CLA adder

19. Calculation of carries – Prefix
Graphs
The components usually seen in a prefix graph are the following:
processing component: buffer component:
(g in1 , pin1 ) ( gin , pin )
( g in2 , pin2 )
( g out , pout ) ( g out , pout )
( g out , pout ) ( g out , pout )
( g out , pout ) = ( g in
1
+ pin 1 ⋅ g in2 , pin1 ⋅ pin 2 ) ( g out , pout ) = ( g in , pin )

20. Prefix graphs for representation of
Prefix addition
 Example: serial adder carry generation represented by prefix graphs
(p8, g8) (p7, g7) (p6, g6) (p5, g5) (p4, g4) (p3, g3) (p2, g2) (p1, g1)
c8 c7 c6 c5 c4 c3 c2 c1

21. Key architectures for carry calculation:
 1960: J. Sklansky – conditional adder
 1973: Kogge-Stone adder
 1980: Ladner-Fisher adder
 1982: Brent-Kung adder
 1987: Han Carlson adder
 1999: S. Knowles
Other parallel adder architectures:
 1981: H. Ling adder
 2001: Beaumont-Smith

22. 1960: J. Sklansky – conditional adder

23. 1960: J. Sklansky – conditional adder
(p8, g8) (p7, g7) (p6, g6) (p5, g5) (p4, g4) (p3, g3) (p2, g2) (p1, g1)
c8 c7 c6 c5 c4 c3 c2 c1
 The Sklansky adder has:
 Minimal depth
 High fan-out nodes

24. 1973: Kogge-Stone adder
(p8, g8) (p7, g7) (p6, g6) (p5, g5) (p4, g4) (p3, g3) (p2, g2) (p1, g1)
c8 c7 c6 c5 c4 c3 c2 c1
 The Kogge-Stone adder has:
 Low depth
 High node count (implies more area).
 Minimal fan-out of 1 at each node (implies faster performance).

25. 1980: Ladner-Fischer adder
(p8, g8) (p7, g7) (p6, g6) (p5, g5) (p4, g4) (p3, g3) (p2, g2) (p1, g1)
c8 c7 c6 c5 c4 c3 c2 c1
 The Ladner-Fischer adder has:
 Low depth
 High fan-out nodes
 This adder topology appears the same as the Schlanskly conditional sum adder. Ladner-Fischer formulated
a parallel prefix network design space which included this minimal depth case. The actual adder they
included as an application to their work had a structure that was slightly different than the above.

26. 1982: Brent-Kung adder
(p8, g8) (p7, g7) (p6, g6) (p5, g5) (p4, g4) (p3, g3) (p2, g2) (p1, g1)
c8 c7 c6 c5 c4 c3 c2 c1
 The Brent-Kung adder is the extreme boundary case of:
 Maximum logic depth in PP adders (implies longer calculation
time).
 Minimum number of nodes (implies minimum area).

27. 1987: Han Carlson adder
 The Han-Carlson adder combines the Brent-Kung and
Kogge-Stone structures into a hybrid structure.
 Efficient
 Suitable for VLSI implementation.

28. 1999: S. Knowles
 Knowles proposed
adders that trade off:
Brent-Kung topology
(Minimum fan-out)  Depth, interconnect,
area.
 These adders are
bound by the
Knowles Lander-Fischer
topologies
(Varied fan-out
(minimum depth)
at each level ) and
Brent-Kung (minimum
Ladner-Fischer fanout) topologies.
topology
(Minimum depth, high
fanout)

29. An interesting taxonomy:
Harris[2003] presented an
interesting 3-D taxonomy of
the adders presented so far.
Each axis represents a
characteristic of the adders:
-Fanout
-Logic depth
-Wire connections
He also proposed the following
structure:

30. 1981: H. Ling adder
Ling Adders are a different family of adders.
They can still be formulated as prefix adders.
Ling adders differ from the “traditional” PP adders in that:
 They are based on a different set of equations.
 The new set of equations introduces the following tradeoffs:
Precalculation of Pi, Gi terms is based on more complex
equations
Calculation of the carries is based
on simpler equations
Final addition stage is more
complex

31. Beaumont-Smith
7 6 5 4 3 2 1 0 G1:0=G1+P1G0
G2:1=G2+P2G1
G2:0=G2+P2G1+P2P1G0
G7:0=G7:4+P7:4G4:0
G3:2=G3+P3G2
6-5 5-4 3-2 2-1 1-0
P7:4=P7P5P4 7-6
G3:1=G3+P3G2+P3P2G1
7-5 6-4 3-1 2-0
G3:0=G3+P3G2+
7-4 3-0
G7:4=G7+P7G6+P7P6G5+P7P6P5G4 P3P2G1+P3P2P1G0
G7:5=G7+P7G6+P7P6G5 G4:0=G4+P4G3+P4P3G2
+P4P3P2G1+P4P3P2P1GO
6-0 5-0 4-0
7-0 G5:4=G5+P5G4
G5:0=G5:4+P5:4G4:0
c8 c7 c6 c5 c4 c3 c2 c1 P5:4=P5P4
G6:5=G6+P6G5
G7:6=G7+P7G6 G6:0=G6+P6G5:4+P6P5:4G4:0 G6:4=G6+P6G5+P6P5G4
P5:4=P5P4

32. Summary (1/3)
 The parallel prefix formulation of binary addition
is a very convenient way to formally describe an
entire family of parallel binary adders.

33. Summary (2/3)
 A parallel prefix adder can be seen as a 3-stage process:
Pre-calculation of Pi, Gi terms
Calculation of the carries.
Simple adder to generate the sum
 There exist various architectures for the carry calculation part.
 Trade-offs in these architectures involve the
 area of the adder
 its depth
 the fan-out of the nodes
 the overall wiring network.

34. Summary (3/3)
 Variations of parallel adders have been
proposed. These variations are based on:
 Modifying the carry generation equations and
reformulating the prefix definition (Ling)
 Restructuring the carry calculation trees based by
optimizing for a specific technology (Beaumond-
Smith)
 Other optimizations.

35. References:
Beaumont-Smith, Cheng-Chew Lim, “Parallel Prefix Adder Design”, IEEE, 2001
Han, Carlson, “Fast Area-Efficient VLSI Adders, IEEE, 1987
Dimitrakopoulos, Nikolos, “High-Speed Parallel-Prefix VLSI Ling Adders”, IEEE 2005
Kogge, Stone, “A Parallel Algorithm for the Efficient solution of a General Class of Recurrence equations”, IEEE, 1973
Simon Knowles, “A Family of adders”, IEEE, 2001
Ladner, Fischer, “Parallel Prefix Computation”, ACM, 1980
Brent, Kung, “A regular Layout for Parallel Adders”, IEEE, 1982
H. Ling, “High-Speed Binary Adder”, IBM J. Res. And Dev., 1980
J. Sklansky, “Conditional-Sum Addition Logic”, IRE transactions on computers, 1960
D. Harris, “A Taxonomy of Parallel Prefix Networks”, IEEE, 2003