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

# Problem statement

The **Catalan** numbers are a special sequence of numbers given by the following set of formulas:


$C_0 = 1$

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

The first expression gives the base case of the formula. The second expression says that the $n^{th}$ Catalan number is simply the sum of products of specific Catalan number pairs. These specific pairs are just Catalan numbers with the same distance from either end of the series.

$C_4$ for example would be equal to:

$C_4 = C_0C_3 + C_1C_2 + C_2C_1 + C_3C_0$

You can already see the hefty amount of recursion in this series.

The Catalan numbers form the following series:
(1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862...)

## Applications of the Catalan numbers

The Catalan numbers readily appear in many interesting counting problems.

- The number of ways to put parentheses around *n* numbers for multiplication.

- The number of paths to climb up a 2n x 2n grid without going above the diagonal.

- The number of possible binary trees with *n* leaf nodes. This has been shown in the visualization below.


In this lesson, you will solve your first coding challenge using bottom-up 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

Challenge: The Catalan Numbers

Problem statement