Examples of Dynamic Programming (DP)
Example 1: Consider the problem of finding the Fibonacci sequence:
Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
Brute Force Approach:
To find the nth Fibonacci number using a brute force approach, you would simply add the (n-1)th and (n-2)th Fibonacci numbers. This would work, but it would be inefficient for large values of n, as it would require calculating all the previous Fibonacci numbers.
Dynamic Programming Approach:
Fibonacci Series using Dynamic Programming
- Subproblems: F(0), F(1), F(2), F(3), …
- Store Solutions: Create a table to store the values of F(n) as they are calculated.
- Build Up Solutions: For F(n), look up F(n-1) and F(n-2) in the table and add them.
- Avoid Redundancy: The table ensures that each subproblem (e.g., F(2)) is solved only once.
By using DP, we can efficiently calculate the Fibonacci sequence without having to recompute subproblems.
Example 2: Longest common subsequence (finding the longest subsequence that is common to two strings)
Example 3: Shortest path in a graph (finding the shortest path between two nodes in a graph)
Example 4: Knapsack problem (finding the maximum value of items that can be placed in a knapsack with a given capacity)
Dynamic Programming or DP
Dynamic Programming is a method used in mathematics and computer science to solve complex problems by breaking them down into simpler subproblems. By solving each subproblem only once and storing the results, it avoids redundant computations, leading to more efficient solutions for a wide range of problems. This article provides a detailed exploration of dynamic programming concepts, illustrated with examples.
Table of Content
- What is Dynamic Programming ?
- How Does Dynamic Programming Work?
- Examples of Dynamic Programming
- When to Use Dynamic Programming?
- Approaches of Dynamic Programming
- Dynamic Programming Algorithm
- Advantages of Dynamic Programming
- Applications of Dynamic Programming
- Learn Basic of Dynamic Programming
- Advanced Concepts in Dynamic Programming
- Dynamic Programming Problems