Introduction: Dynamic programming (DP) is a powerful technique used to solve problems by breaking them down into simpler subproblems. It's widely used in competitive programming to optimize solutions that have overlapping subproblems and optimal substructure.

What is Dynamic Programming? Dynamic programming is essentially recursion with memoization. The key idea is to solve a problem by solving its subproblems, storing the results of subproblems to avoid redundant work, and building up the solution to the overall problem from these subproblem solutions.

When to Use Dynamic Programming? DP is particularly useful when a problem can be divided into smaller overlapping subproblems. Problems that require optimal decisions or have a recursive structure are prime candidates.

Basic Steps:

Define the State: Determine what the subproblem is and how to represent it. State Transition: Identify how the solution to one subproblem can be built from the solutions to other subproblems. Base Cases: Establish the simplest subproblems that can be solved directly. Memoization: Store the results of subproblems to avoid recalculating them. Example Problem: Fibonacci Sequence

Recursive Solution:

def fib(n): if n <= 1: return n return fib(n-1) + fib(n-2)

DP Solution:

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]

Common DP Problems:

Longest Common Subsequence Knapsack Problem Coin Change Problem

Conclusion: Dynamic programming is a crucial concept in competitive programming. By mastering it, you can solve many problems efficiently. Start with simple problems and gradually work your way up to more complex ones.