**Tom Kelliher, CS 240**

**Feb. 20, 2012**

Decoders, encoders, muxes.

- A ``fast binary adder.''
- The lower bound for addition speed.

Carry lookahead addition, other integer representations.

- Ripple carry adder: example of reuse and divide and conquer.
- Wire together full binary adders in order to add two bit
numbers.
- Running time of a ripple carry adder.
Running time of a full binary adder is O. Ripple carry adder?

**Excessive!!!**

- Wire together full binary adders in order to add two bit
numbers.
- An O (!!!) bit adder.
- Important equations (briefly explain):
Carry generate at bit position : .

Carry propagate at position : .

- Carry-in is .
.

.

.

Etc.

- What's the circuit depth of ?
- What's wrong with this picture?

- Important equations (briefly explain):

- Due to Winograd.
- Our abstract gate for an idealized adder: the gate
- Independent of technology.
- is the fan-in.
- is the radix (binary).
- Assume that such a gate can compute any -valued function of
inputs.

- Independent of technology.
- Adding bit numbers. How many inputs? How many outputs? What is
the
*minimum*number of inputs some output is dependent upon? The*maximum*? - For the output dependent upon that maximum, consider the ideal
circuit which reduces its inputs to the final output.
- What does it look like?
- How many intermediate signals remain after one level of gates?
- What is its depth? -- That's the lower bound for addition.

- What does it look like?

Thomas P. Kelliher 2012-02-18 Tom Kelliher