Gain insights into algorithms and their implementation for problem-solving in Python. Learn about recursion, dynamic programming, greedy algorithms, and graph algorithms to enhance coding proficiency and problem-solving skills.

As a developer, mastering the concepts of algorithms and being proficient in implementing them is essential to improving problem-solving skills. This course aims to equip you with an in-depth understanding of algorithms and how they can be utilized for problem-solving in Python.

Starting with the basics, you'll gain a foundational understanding of what algorithms are, with topics ranging from simple multiplication algorithms to analyzing algorithms. Then, you’ll delve into more advanced topics like recursion, backtracking, dynamic programming, and greedy algorithms. You'll also learn about basic graph algorithms, depth-first search, shortest paths, minimum spanning trees, and all-pairs shortest paths. 

By the end of this course, you'll have acquired a wide range of skills that will significantly enhance your ability to solve problems efficiently in Python. Through this course, not only will you improve your coding skills, but you will also gain confidence in tackling complex problems.

Mastering Algorithms for Problem Solving in Python

# Understanding the game constraints
Consider the following simple two-player game played on an $n × n$ square grid with a border of squares; let’s call the players Horace Fahlberg-Remsen and Vera Rebaudi. Each player has $n$ tokens that they move across the board from one side to the other. Horace’s tokens start in the left border, one in each row, and move _horizontally_ to the right; symmetrically, Vera’s tokens start in the top border, one in each column, and move _vertically_ downward. The players alternate turns. In each of his turns, Horace either moves one of his tokens one step to the right into an empty square or jumps one of his tokens over exactly one of Vera’s tokens into an empty square two steps to the right. If no legal moves or jumps are available, Horace simply passes. Similarly, Vera either moves or jumps one of her tokens downward in each of her turns unless no moves or jumps are possible. The first player to move all their tokens off the edge of the board wins. (It’s not hard to prove that as long as there are tokens on the board, at least one player can legally move, and therefore someone eventually wins.)

Unless we’ve seen this game before, we probably don’t have any idea how to play it well. Nevertheless, there is a relatively simple backtracking algorithm that can play this game—or any two-player game without randomness or hidden information that ends after a finite number of moves—*perfectly*. That is, if we drop into the middle of a game, and it is possible to win against another perfect player, the algorithm will tell us how to win.
# What is state?
A **state** of the game consists of the locations of all the pieces and the identity of the current player. These states can be connected into a **game tree**, which has an edge from state x to state y if and only if the current player in state x can legally move to state y. The root of the game tree is the initial position of the game, and every path from the root to a leaf is a complete game.

# Good and bad game states 

To navigate through this game tree, we recursively define a game state to
be "good" or "bad" as follows:
- A game state is good if either the current player has already won or if the current player can move to a bad state for the opposing player.
- A game state is bad if either the current player has already lost or if every available move leads to a good state for the opposing player.


Equivalently, a non-leaf node in the game tree is good if it has at least one bad child, and a non-leaf node is bad if all its children are good. By induction, any player that finds the game in a good state on their turn can win the game, even if their opponent plays perfectly; on the other hand, starting from a bad state, a player can win only if their opponent makes a mistake. This recursive definition was proposed by Ernst Zermelo in 1913.

This recursive definition immediately suggests the following recursive back-tracking algorithm to determine whether a given game state is good or bad. At its core, this algorithm is just a depth-first search of the game tree; equivalently, the game tree is the recursion tree of the algorithm! A simple modification of this backtracking algorithm finds a good move (or even all possible good moves) if the input is in a good game state.

$\underline{PlayAnyGame(X , player):} 
\\ \hspace{0.4cm}
if\space player\space has\space already\space won\space in\space state\space X
\\ \hspace{1cm}
return\space Good
\\ \hspace{0.4cm}
if\space player\space has\space already\space lost\space in\space state\space X
\\ \hspace{1cm}
return\space Bad
\\ \hspace{0.4cm}
for\space all\space legal\space moves\space X \rightarrow  Y
\\ \hspace{1cm}
if\space PlayAnyGame(Y, ¬player) = Bad
\\ \hspace{1.7cm}
return\space Good \hspace{1cm} {\color{Red} 〈〈\space X \rightarrow Y\space is\space a\space good\space move\space  〉〉}
\\ \hspace{0.4cm}
return\space Bad  \hspace{2.35cm} {\color{Red} 〈〈\space 
 There\space are \space no\space good\space moves\space 〉〉}
$
   

Learn about the different techniques used to solve problems using game trees.

Getting Started

Introduction to Algorithm

Recursion

Backtracking

Dynamic Programming

Greedy Algorithms

Prove Your Skills: A Five-Chapter Assessment

Basic Graph Algorithms

Depth-First Search

Minimum Spanning Trees

Shortest Paths

All-Pairs Shortest Paths

Pushing Your Limits: A Comprehensive Assessment

Wrapping up

Game Trees

Understanding the game constraints