Dynamic Programming (DP) can be challenging, but following a structured approach makes it easier to solve problems efficiently. Here’s a step-by-step guide to mastering DP:

1. Recognize That It's a DP Problem

• Overlapping Subproblems → The problem can be broken into smaller subproblems that repeat.
• Optimal Substructure → The solution to the overall problem can be built using solutions to subproblems.
• Common DP Problem Types:
  • Fibonacci sequence
  • Knapsack problem
  • Longest common subsequence
  • Coin change problem

2. Define the State (DP Array or Function)

• Identify what dp[i] represents in the problem.
• Example: In Fibonacci, dp[i] = i-th Fibonacci number.
• Example: In the Knapsack problem, dp[i][j] = maximum value for the first i items with j weight capacity.

3. Formulate the Recurrence Relation

• Express the current state in terms of previous states.
• Example (Fibonacci): $$dp[i] = dp[i-1] + dp[i-2]$$
• Example (Knapsack): $$dp[i][j] = \max(dp[i-1][j], \text{value}[i] + dp[i-1][j - \text{weight}[i]])$$
• Tip: Think about the choices you can make at each step.

4. Decide on Bottom-Up (Tabulation) or Top-Down (Memoization) Approach

• Top-Down (Recursion + Memoization)
  • Write a recursive function and store computed values in a hash table or array.
  • Example:

    python
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    def fib(n, memo={}):     if n <= 1:         return n     if n not in memo:         memo[n] = fib(n-1, memo) + fib(n-2, memo)     return memo[n] 

• Bottom-Up (Tabulation)
  • Use an iterative approach to fill an array.
  • Example:

    python
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    def fib(n):     dp = [0] * (n + 1)     dp[1] = 1     for i in range(2, n + 1):         dp[i] = dp[i-1] + dp[i-2]     return dp[n] 

5. Optimize Space Complexity

• Instead of storing all states, keep track of only what’s necessary.
• Example (Optimized Fibonacci using two variables):

  python
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  def fib(n):     if n <= 1:         return n     prev2, prev1 = 0, 1     for _ in range(2, n + 1):         curr = prev1 + prev2         prev2, prev1 = prev1, curr     return prev1 

• This reduces O(n) space to O(1).

6. Practice Common DP Patterns

Master these common DP patterns:

1. 1D DP → Fibonacci, Climbing Stairs
2. 2D DP → Longest Common Subsequence, Knapsack
3. Subsequence DP → Palindromic Substrings, Edit Distance
4. Graph-based DP → Shortest Paths (Floyd-Warshall, Bellman-Ford)
5. Bitmask DP → Traveling Salesman Problem

7. Solve Problems Step by Step

• Start with Easy DP Problems (Fibonacci, Climbing Stairs).
• Move to Medium Problems (Knapsack, Longest Common Subsequence).
• Tackle Hard Problems (Subset Sum, Word Break, DP on Trees).