Learn about dynamic programming in Python, delve into recursion basics, explore advanced DP techniques, and discover practical coding challenges to optimize algorithms for real-world applications.

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

You are given an NxN chessboard, and you are required to place N queens on this chessboard such that no queen is under threat from any other queen. 

> In chess a queen can move any number of steps horizontally, vertically, or diagonally.

This means that no queen should share a row, column, or diagonal with another queen.


## Input 

As input, your function will take a number `n`, which is the size of the board, and a 2-D list of strings as `board`, which is a grid where each row is a list of strings. Each string represents a cell on the board, initially set to '-' to show that the cell is empty.

```
n = 4
board = [["-", "-", "-", "-"], 
         ["-", "-", "-", "-"],
         ["-", "-", "-", "-"],
         ["-", "-", "-", "-"]]
```

## Output

Your function should return the updated `board`, such that no queen denoted by `q` shares a row, column, or diagonal with another queen.

```
placeNQueens(n, board) = 
       [["-", "q", "-", "-"],
        ["-", "-", "-", "q"],
        ["q", "-", "-", "-"],
        ["-", "-", "q", "-"]]
```

# Coding challenge
You can use the `isSafe(i, j, board)` function to test whether the position given by `i`$^{th}$ row and `j`$^{th}$ column is safe to place a queen. This function basically checks whether the box position given by row `i` and column `j` shares row, column, or diagonal with any other queen you had already placed on the `board`.

Do not change the prototype of either function as these are being used for testing. You can make your own helper functions though.

This one is a little trickier, but at the same time, it unleashes the real power of recursion. Feel free to look at hints. Best of luck!




In this lesson, you will be challenged with a classic recursion problem, placing N queens on an NxN chessboard.


Challenge: Place N Queens on an NxN Chessboard

In this lesson, you will be challenged with a classic recursion problem, placing N queens on an NxN chessboard.

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

Solving Sudoku and the 8-Queens Puzzle as Constraint Satisfaction

Challenge: Place N Queens on an NxN Chessboard

Problem statement

Input