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Dynamic Programming

Dynamic Programming (DP) is a method for solving complex problems by breaking them down into simpler subproblems. It's one of the most important techniques in competitive programming and algorithmic problem-solving.

🎯 What is Dynamic Programming?​

Dynamic Programming is an optimization technique that solves problems by:

  1. Breaking down complex problems into simpler subproblems
  2. Storing solutions to subproblems to avoid redundant calculations
  3. Building up the solution to the original problem from subproblem solutions

📚 Key Concepts​

1. Optimal Substructure​

A problem has optimal substructure if an optimal solution contains optimal solutions to subproblems.

2. Overlapping Subproblems​

The same subproblems are solved multiple times. DP stores these solutions to avoid recalculating.

3. Memoization vs Tabulation​

  • Memoization: Top-down approach (recursive with caching)
  • Tabulation: Bottom-up approach (iterative)

🚀 Common DP Patterns​

1. 1D Dynamic Programming​

// Example: Fibonacci
int dp[n+1];
dp[0] = 0, dp[1] = 1;
for(int i = 2; i <= n; i++) {
    dp[i] = dp[i-1] + dp[i-2];
}

2. 2D Dynamic Programming​

// Example: Grid DP
int dp[n][m];
// Fill based on problem requirements

3. Knapsack Problems​

  • 0/1 Knapsack
  • Unbounded Knapsack
  • Fractional Knapsack

4. Longest Common Subsequence (LCS)​

int dp[n+1][m+1];
// dp[i][j] = length of LCS of first i chars of s1 and first j chars of s2

📝 Problem List​

Basic DP​

Grid DP​

Advanced DP​

💡 Tips for DP Problems​

  1. Identify the State: What information do you need to represent a subproblem?
  2. Define the Transition: How do you move from one state to another?
  3. Set Base Cases: What are the smallest subproblems?
  4. Choose Implementation: Top-down (memoization) or bottom-up (tabulation)?
  5. Optimize Space: Can you reduce space complexity?

🔗 Related Resources​

Master DP and unlock the power to solve complex algorithmic problems efficiently! 🚀