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

def catalan(n):
  if n == 0:      # base case; C(0) = 1
    return 1
  sum = 0
  # iterate from 1...n to evaluate: C(0)*C(n-1) + C(1)*C(n-2) ... + C(n-1)*C(0)
  for i in range(n):  
    sum += (catalan(i) * catalan(n-1-i))  # C(i)*C(n-1-i)
  return sum

print(catalan(4))

## Explanation

This is a much simpler problem. The only thing we need to figure out is how to convert the following summation equation into code:


$C_n = \sum_{i=0}^{n-1} C_iC_{n-1-i}$

or

$C_n = C_0C_{n-1} + C_1C_{n-2} + C_2C_{n-3} + ... + C_{n-1}C_0$

So, for the sum, we simply use a for loop to iterate from `0` to `n-1`, and then we make calls to `catalan(i)` and `catalan(n-1-i)` as required by the expression (*lines 6-7*). The following is our base case:


$C_0 = 1$

This has been handled in *lines 2 and 3*.

Although the repeating subproblems are very apparent in this problem, let's still look at a dry run of the algorithm to get a better hang of things.



In this lesson, we will review some solutions to the Catalan numbers challenge from the last lesson.


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 Catalan Numbers

Solution 1: Simple recursion #

Explanation