Carry Lookahead Addition

Tom Kelliher, CS 315

Feb. 26, 1999



Homework due Monday.


Read 1.2.2.

From Last Time

Summary of fixed-connection network model.


  1. Carry lookahead addition algorithm, proof.

Coming Up

Parallel prefix computations.

Carry Lookahead Addition

  1. In general, the sum of two n-bit numbers results in an n+1-bit sum.

  2. The problem reduces to computing the carries.

  3. For each bit position we determine if it would
    1. Propagate a carry (p).

    2. Generate a carry (g).

    3. Stop a carry (s).

    What's the ``truth table'' for this.

  4. Example: 0x5c92 and 0x685c.

  5. The computation is carried out in a binary tree with addend, augend bits transmitted to the leaves.

    Dimensions of the tree?

  6. Steps of the computations:
    1. Step 1: compute p, g, s in the leaves.

    2. ascending steps:
      1. Receive inputs from children, who ``erase'' their values.

      2. Pass value received from right child to left child.

      3. Compute own value to pass to parent. What is the computation?

    3. Special case for root:
      1. Its value becomes the carry-out (or additional sum bit). Output this value.

      2. Set value to s.

    4. descending steps:
      1. Receive value from parent.

      2. Keep first non-p value received (including the value received by a left child from a right child).

  7. When finished, all leaves are labeled either s or g. Which is a carry-in of 0 and which is 1?

  8. We know no leaf will be ``stuck'' with a p. Why?

  9. The carry-in for a leaf is the left-most non-p value to the right of the leaf, as originally computed by the leaves.

    Prove that the algorithm stores this carry in each leaf.

Thomas P. Kelliher
Fri Feb 26 10:24:32 EST 1999
Tom Kelliher