Carry-Save Addition

Tom Kelliher, CS 315

Apr. 5, 1999




Read 1.2.4.

From Last Time

Segmented prefix problem.


  1. Brief review of carry lookahead addition.

  2. Carry-Save addition.

Coming Up

Multiplication and convolution.

Previous Results

Carrylookahead addition:

  1. Add two k-bit numbers in bit steps.

  2. Suppose we wish to add N k-bit numbers as quickly as possible.
    1. What is the organization of the adder?

    2. Do the adders become wider as we progress?

    3. Running time:

      Are these really equivalent?

      Can we do better?

Carry-Save Addition

  1. Idea: Add three k-bit numbers in a single bit step, producing two k+1-bit numbers.

  2. Implementation: Use a binary full adder for each bit position, producing a low order bit and a carry bit for each bit position.

  3. The final sum is the sum of the low order bits and the shifted carry bits.

  4. Example:

  5. Time required to reduce the sum of N k-bit numbers to the sum of two -bit numbers: bit steps.


    1. After the first step of CSA, how many numbers are we left with? (At most . Why?)

    2. After the second step?

    3. After the jth step:

    4. After steps, we're left with numbers, so we need steps to reduce to two -bit numbers.

  6. Use a carry lookahead adder to reduce the two-number redundant representation to a single, non-redundant number.

  7. Example: Draw the Wallace tree for N = 9.

Thomas P. Kelliher
Mon Apr 5 11:02:53 EDT 1999
Tom Kelliher