Worked Examples: Dynamic Programming I

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

Example 1: Fibonacci

❓ Problem: Naive vs DP complexity.

💡 Solution: Naive $O(2^n)$. DP $O(n)$.

Example 2: Rod Cutting Recursive

❓ Problem: Recurrence?

💡 Solution: $r_n = \max(p_i + r_{n-i})$.

Example 3: Memoization

❓ Problem: What do we store?

💡 Solution: Max price for length $i$ in array $R[i]$.

Example 4: Reconstruction

❓ Problem: How to find the cuts?

💡 Solution: Store choice $S[i]$ (first cut). Backtrack.

Example 5: Subproblems

❓ Problem: Count for Rod Cutting $n$?

💡 Solution: $n+1$ subproblems.

Example 6: Space

❓ Problem: Space complexity?

💡 Solution: $O(n)$ for 1D table.