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

# Task
There are $n$ galaxies connected by $m$ intergalactic teleport-ways. Each
teleport-way joins two galaxies and can be traversed in both directions. Also,
each teleport-way $e$ has an associated cost of $c(e)$ dollars, where $c(e)$ is a
positive integer. A teleport-way can be used multiple times, but the toll must
be paid every time it is used.
Judy wants to travel from galaxy $s$ to galaxy $t$, but teleportation is not
very pleasant and she would like to minimize the number of times she needs
to teleport. However, she wants the total cost to be a multiple of five dollars,
because carrying small change is not pleasant either.

Analyze the given algorithm to compute the smallest number of
times Judy needs to teleport to travel from galaxy $s$ to galaxy $t$ so that
the total cost is a multiple of five dollars.

# Logic building
To find the smallest number of times Judy needs to teleport from galaxy $s$ to galaxy $t$, so that the total cost is a multiple of five dollars, we can use a modified version of the Dijkstra's algorithm. We will use $tuple (cost, count)$ to store the cost of teleporting and the count of teleports. The cost will be stored as a pair of integers: the cost modulo 5 and the actual cost.

**Algorithm**

- Create a dictionary $dist$ to store the minimum teleport count and cost for each galaxy. Initialize the distance from $s$ to itself as $(0, 0)$ and the distance to all other galaxies as ($float('inf'), float('inf'))$.
- Create a min heap $priority_-queue$ containing a tuple $(cost_-mod_-5, cost, count, galaxy)$ for each galaxy, initialized with $(0, 0, 0, s)$.
- While the $priority_-queue$ is not empty, pop the smallest element (based on the teleport count) from it. Let the popped element be $(curr_-cost_-mod_-5, curr_-cost, curr_-count, curr_-galaxy).$
- If $curr_-galaxy$ is equal to $t$, return $curr_-count$ if $curr_-cost_-mod_-5 == 0$. Otherwise, continue to the next step.
- For each neighboring galaxy $next_-galaxy$ and its teleport-way cost $next_-cost$:
     - Calculate the new cost and count like so: $new_-cost = curr_-cost + next_-cost$ and $new_-count = curr_-count + 1$.
     - Calculate the new cost modulo 5 as $new_-cost_-mod_-5 = new_-cost % 5$.
      
   - If the new teleport count is smaller than the previously stored count for $next_-galaxy$, update the tuple $(new_-cost_-mod_-5, new_-cost, new_-count)$ in $dist[next_-galaxy]$ and push the updated tuple $(new_-cost_-mod_-5, new_-cost, new_-count, next_-galaxy)$ into the priority queue.
- If the loop finishes and no solution is found, return $-1$, meaning there is no valid path with the total cost being a multiple of five dollars.

# Time complexity
The time complexity of this algorithm is $O((m + n) \log n)$, which is similar to the time complexity of Dijkstra's algorithm with a priority queue.

# Solution
Following is the solution to the above challenge:

Solution to the problem related to basic graph algorithms.

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

Solution: Basic Graph Algorithms

Task