# Divide and Conquer - Thu, January 21

## Master Theorem for Recurrences

In general, , where . The Master Theorem states the following:

We need to bound our generic total-work formula, in the three cases above. Recall that for a power series, .

### Case 1: Root-Heavy

If , then , because all the other terms are constants.

If , then .

If , then .

## Matrix Multiplication

We'll use input matrices and of size x each. In general, matrix multiplication is not commutative.

### The Naive Algorithm

For i = 1, ..., n:      n times
For j = 1, ..., n:	n times
compute Z_ij	O(n)

This algorithm runs in , which is pretty bad.

### The Recursive Algorithm

We can leverage blockwise multiplication.

Here, , because there are eight multiplications and the recombination cost (summing two x matrices takes time) is . Applying the third case of Master Theorem will tell us that our new algorithm is... also .

### Beyond Strassen

Suppose you figure out how to multiply x matrices in multiplications. Then, via divide and conquer, you get an algorithm in time. This boils down to . We can hence "lift" any finite-size improvement into an asymptotic improvement.

Currently, the fastest matrix multiplication algorithm runs in .

## Recap of Examples

We have now seen

• Karatsuba, which is faster integer multiplication ()
• Strassen, which is faster matrix multiplication ()