Worked Examples: Course Review

Practice these representative problems to check your understanding of each paradigm.

1. Asymptotic Analysis

❓ Compare the following: $n!$ vs $2^n$ vs $n^3$.

💡 $n^3 = O(2^n)$ and $2^n = O(n!)$. Factorial grows faster than exponential, which grows faster than polynomial.

2. Master Theorem

❓ Solve $T(n) = 2T(n/2) + n$.

💡 $a=2, b=2, f(n)=n$. $n^{\log_b a} = n^{\log_2 2} = n^1$. Since $f(n) = \Theta(n)$, this is Case 2. $T(n) = \Theta(n \lg n)$.

3. Greedy Choice

❓ Why does a greedy approach work for fractional knapsack but not 0/1?

💡 Fractional allows local optimization per unit weight (Greedy). 0/1 requires considering all combinations because items are indivisible (needs DP).

4. Dynamic Programming (LCS)

❓ What is the LCS of "ABCBDAB" and "BDCABA"?

💡 Length is 4. One possible LCS is "BDAB". Use the table $c[i,j] = c[i-1,j-1]+1$ if match, else $\max(c[i,j-1], c[i-1,j])$.

5. Graph Pathfinding

❓ When would you choose Bellman-Ford over Dijkstra?

💡 When the graph contains negative edge weights or when you need to detect negative weight cycles.

6. NP-Completeness

❓ If $P = NP$, how does it affect cryptography?

💡 If $P=NP$, most encryption schemes (like RSA) could be broken easily because "hard" problems like factoring would have polynomial solutions.