Lecture 10: Karatsuba Multiplication & Master Theorem Intro [11/02/2026]

Course: Advanced Data Structures and Algorithms

1. The Integer Multiplication Problem [00:01:14]

High School Method: Multiplying two $n$-digit numbers digit-by-digit requires $n^2$ multiplications and $n^2$ additions ($\Theta(n^2)$).
Kolmogorov's Conjecture: Andrey Kolmogorov conjectured in conferences that $O(n^2)$ was optimal for integer multiplication.
Divide and Conquer (Naive) [00:04:37]:
- Split $A$ into $A_1$ (first $n/2$ bits) and $A_2$ (last $n/2$ bits): $A = A_1 2^{n/2} + A_2$.
- $A \cdot B = (A_1 2^{n/2} + A_2)(B_1 2^{n/2} + B_2)$
- $A \cdot B = A_1 B_1 2^n + (A_1 B_2 + A_2 B_1) 2^{n/2} + A_2 B_2$
- This requires 4 multiplications of size $n/2$.
- Recurrence: $T(n) = 4T(n/2) + O(n)$.
- Result: Solving this gives $T(n) = O(n^2)$, which fails to improve on the naive method.

The Trick [00:36:25]: Anatoly Karatsuba, a graduate student, realized we could compute the middle term $(A_1 B_2 + A_2 B_1)$ using only one additional multiplication:
- Let $M_1 = A_1 B_1$
- Let $M_2 = A_2 B_2$
- Let $M_3 = (A_1 + A_2)(B_1 + B_2)$
- Then $(A_1 B_2 + A_2 B_1) = M_3 - M_1 - M_2$.
New Recurrence [00:39:30]: $$T(n) = 3T(n/2) + O(n)$$
Complexity Analysis [00:46:57]:
- Solving the recurrence leads to $T(n) \approx 15 \cdot 3^{\log_2 n}$.
- Using the log property $a^{\log_c b} = b^{\log_c a}$, we get $n^{\log_2 3}$.
- $\log_2 3 \approx 1.585$.
- Final Complexity: $O(n^{1.59})$, a significant improvement over $O(n^2)$.

The Master Form: $T(n) = a T(n/b) + n^c$.
Parameters:
- $a$: Number of subproblems.
- $b$: Factor by which subproblem size is reduced.
- $n^c$: Combining/Dividing cost.
Teaser: The instructor will derive the general solution (Master Theorem) in the next lecture to avoid manual recurrence solving every time.

Closest Pair Problem: Find the two closest points among $n$ points in a 2D plane.
Naive Complexity: $O(n^2)$ by checking all $n(n-1)/2$ pairs.
Challenge [01:09:22]: Can we do it in less than $O(n^2)$? The instructor encourages students to struggle with this for a few days to appreciate the forthcoming D&C solution ($O(n \log n)$).
Philosophy [01:10:25]: "Only if you wander in the hot sun will you realize the value of the shade." Struggle is necessary to appreciate creative algorithm design.

Key Terms Corrected: