Dynamic Programming (DP) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems. It is applicable where the problem can be broken down into overlapping subproblems that can be solved independently.
Basic Concepts
- Optimal Substructure: A problem exhibits optimal substructure if an optimal solution to the given problem can be constructed efficiently from optimal solutions of its subproblems.
- Overlapping Subproblems: When a problem can be broken down into overlapping subproblems, then the problem has overlapping subproblems.
Types of Dynamic Programming
- Memoization: Store the results of expensive function calls and re-use them when the same inputs occur again.
- Tabulation: Build up the solution iteratively and store the results of subproblems to avoid redundant calculations.
Example Problem: Fibonacci Sequence
The Fibonacci sequence is a classic example of a problem that can be solved using dynamic programming.
- Recursive Approach: Exponential time complexity.
- Dynamic Programming Approach: Linear time complexity.
def fibonacci(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]
Resources
For more information and advanced topics in dynamic programming, check out our Advanced Dynamic Programming Tutorial.
Conclusion
Dynamic Programming is a powerful technique that can solve complex problems efficiently. By breaking down problems into smaller subproblems and storing their solutions, we can optimize our algorithms.
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