Parallel Prefix Computations

Tom Kelliher, CS 315

Mar. 3, 1999

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Assignment

Read 1.2.3.

From Last Time

Carry lookahead addition.

Outline

1. Introduction.

2. Computing the prefixes.

Coming Up

Carry save addition.

Parallel Prefix Computations

Introduction

1. General form of a prefix computation:

   

   where is a function of an input bit.

2. Consider carry lookahead addition. Let be the s, p, or g for bit position i. Then:

   

   Note that this is written ``backwards.''

   Why the initial s?

3. Multiplication table for carry lookahead addition:

   

   This operation must be associative for the carry computation to work. Why? Show.

   What's going on here?

Computing the Prefixes

1. Using a four-leaf tree, label each leaf with it's and values.

2. Can the class derive the calculation for the tree? Hint: each vertex in the tree stores the product of its sub-tree leaves.

3. General idea:
   1. Rising phase: each vertex computes the product of inputs from its children. Action of a leaf, internal vertex.

   2. Computed values are passed from left child to right child.

   3. Falling phase: each internal vertex takes received value and transmits it to children. Leaves concatenate received values on left.

4. Show an example on a nine-leaf tree.