Worked Examples: Divide and Conquer

These examples are designed to step through the logical process of applying the algorithmic concepts.

Example 1: Merge Sort Recurrence

❓ Problem: Solve $T(n) = 2T(n/2) + n$.

💡 Solution: $O(n \log n)$.

Example 2: Binary Search Recurrence

❓ Problem: Solve $T(n) = T(n/2) + 1$.

💡 Solution: $O(\log n)$.

Example 3: Bad Split

❓ Problem: Solve $T(n) = T(n-1) + n$ (e.g. Selection Sort).

💡 Solution: $O(n^2)$.

Example 4: Ternary Search

❓ Problem: Split into 3 parts, look in 1. $T(n) = T(n/3) + 1$.

💡 Solution: $O(\log_3 n) = O(\log n)$.

Example 5: Max Element D&C

❓ Problem: Find max of array. $T(n)=2T(n/2)+1$.

💡 Solution: $O(n)$. No improvement over iterative.

Example 6: Stooge Sort

❓ Problem: Weird sort. $T(n) = 3T(2n/3) + 1$.

💡 Solution: Apply Master Theorem (Case 1).