Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable where the problem can be broken down into overlapping subproblems that can be solved independently.

Basics of Dynamic Programming

  1. Optimal Substructure: The optimal solution to the given problem can be constructed efficiently from optimal solutions of its subproblems.
  2. ** overlapping subproblems**: The space of subproblems must be small, so that a recursive algorithm will be efficient.
  3. Memoization: Save the results of expensive function calls and return the cached result when the same inputs occur again.

Common Dynamic Programming Problems

  • Fibonacci Sequence
  • Knapsack Problem
  • Longest Common Subsequence
  • Matrix Chain Multiplication

Example: Fibonacci Sequence

The Fibonacci sequence is a series of numbers where the next number is the sum of the two preceding ones. The sequence starts from 0 and 1.

  • 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

To compute the nth Fibonacci number using dynamic programming, we can use the following formula:

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

Further Reading

For more in-depth tutorials and examples, check out our Dynamic Programming Deep Dive.

Dynamic Programming Deep Dive