2D DP (0/1 Knapsack, Unique Paths, Edit Distance, MCM)

2D Grid DP Patterns

0/1 Knapsack: $dp[i][w] = \max(dp[i-1][w], val[i-1] + dp[i-1][w - wt[i-1]])$.

Production Code Example:

public class EditDistance {
    public int minDistance(String w1, String w2) {
        int m = w1.length(), n = w2.length();
        int[][] dp = new int[m + 1][n + 1];
        for (int i = 0; i <= m; i++) dp[i][0] = i;
        for (int j = 0; j <= n; j++) dp[0][j] = j;
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
                if (w1.charAt(i - 1) == w2.charAt(j - 1)) dp[i][j] = dp[i - 1][j - 1];
                else dp[i][j] = 1 + Math.min(dp[i - 1][j - 1], Math.min(dp[i - 1][j], dp[i][j - 1]));
            }
        }
        return dp[m][n];
    }
}

Key Complexity & Algorithmic Takeaways:

When implementing 2D DP (0/1 Knapsack, Unique Paths, Edit Distance, MCM) in coding interviews and production applications, keep these core guidelines in mind:

• Time Complexity Analysis: Always evaluate best-case, average-case, and worst-case time complexities ($O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$).
• Space Complexity & Memory Bounds: Account for auxiliary memory usage, call stack frame recursion overhead, and heap allocations.
• Edge Cases & Validation: Test empty inputs, null pointers, single-element collections, duplicate values, and integer overflow bounds.
• Optimal vs Naive Solutions: Start with a clear brute-force solution, then optimize using techniques like Hashing, Two Pointers, Windowing, or Dynamic Programming.

Summary Takeaway:

Mastering 2D DP (0/1 Knapsack, Unique Paths, Edit Distance, MCM) provides the foundational problem-solving skills needed to pass technical coding interviews at top tech companies and write ultra-performant software systems.