Gain insights into dynamic programming, explore recursion basics, and delve into advanced techniques like Bottom-Up optimization. Discover ways to solve complex problems more efficiently with hands-on coding challenges.

Dynamic programming is something every developer should have in their toolkit. It allows you to optimize your algorithm with respect to time and space — a very important concept in real-world applications.

In this course, you’ll start by learning the basics of recursion and work your way to more advanced DP concepts like Bottom-Up optimization. Throughout this course, you will learn various types of DP techniques for solving even the most complex problems. Each section is complete with coding challenges of varying difficulty so you can practice a wide range of problems.

By the time you’ve completed this course, you will be able to utilize dynamic programming in your own projects.

Dynamic Programming in Python: Optimizing Programs for Efficiency

import numpy as np
def minMultiplications(dims):
    if len(dims) <= 2:
        return 0
    minimum = np.inf
    for i in range(1,len(dims)-1):
        minimum = min(minimum, minMultiplications(dims[0:i+1]) + minMultiplications(dims[i:]) +
                    dims[0] * dims[-1] * dims[i])
    return minimum

print(minMultiplications([3, 3, 2, 1, 2]))

## Explanation

We can easily imagine `dims` to be a list of matrices. Then each recursive call tries to find the optimal placement of two parentheses between these matrices. So, let's say we have the following sequence of matrices: $A_1 A_2 A_3A_4$. Each recursive call tries to place two parentheses, so we have the following possibilities:

- $(A_1)(A_2A_3A_4)$
- $(A_1A_2)(A_3A_4)$
- $(A_1A_2A_3)(A_4)$

Thus, we make recursive calls for all these possibilities and finally choose the best one from among them.

Look at the following visualization for more clarification.


In this lesson, we will solve the matrix chain multiplication problem with different techniques of dynamic programming.

Chapter 1: From Recursion to Dynamic Programming

Chapter 2: Top-Down Dynamic Programming with Memoization

Chapter 3: Bottom-Up Dynamic Programming with Tabulation

Chapter 4: Practice Problems

Solution Review: The Matrix Chain Multiplication

Solution 1: Simple recursion #

Explanation