The document discusses the Master Theorem and its components (a, b, d), illustrated through examples of divide-and-conquer multiplication of large numbers. It outlines the operations involved, including digit shifts and the carry process for addition and subtraction. Additionally, it describes recursion trees for different multiplication approaches and their corresponding time complexities.

1. Comments from last time
• Purpose of Master theorem?
• What does a, b, and d represent
• Finding a, b, d from pseudocode

2. Div-Conquer Multiplication
1234 × 5678
n: lenth of numbers,
input size: 2n
12 × 78
1×5
34 × 56 34 × 78
12 × 56
1×6 2×5 2×6

3. O(n) time operations
n: lenth of numbers
Shift  multiply by 10: take
array, move all elements
Addition / subtraction:
add/subtract digits, then carry

4. Comment cards
https://goo.gl/RKd8vq

5. Faster Div-Conquer Multiplication
1234 × 5678
34 × 78 (12 + 56) × (34 + 78)
12 × 56
=
68 × 112

6. Faster Div-Conquer Multiplication
12345678
× 87654321
5678
× 4321
(1234 + 5678)
× (8765 + 4321)
1234
× 8765
=
6912
× 13086

7. Recursion Tree: T(n) = 3T(n / 2) + O(n)
n: lenth of numbers,
input size: 2n
n
n/2 n/2 n/2
n/4
n/4 n/4 n/4 n/4 n/
n/4
/4 n/4

8. Recursion Tree: T(n) = 2T(n / 3) + O(n)
n
n/3
n/9 n/9 n/9 n/9
n/3

9. Recursion Tree: T(n) = 3T(n / 3) + O(n)
n/3
n/9 n/9 n/9 n/9 n/9 n/9
n/3
n
n/9 n/9 n/9
n/3