Cheapest Flights Within K Stops 🎯

Welcome to this exciting lesson on Data Structures and Algorithms where we'll dive into the topic of finding the cheapest flights within K stops! This lesson is designed for both beginners and intermediates, so let's get started! 📝

What are we trying to achieve? 💡

In this problem, we are given a list of flights between cities and their costs. We need to find the minimum cost to travel from one city (src) to another (dest) with at most K stops. Let's break it down!

Understanding the Problem 📝

n cities connected by m flights
Each flight has a cost and an associated number of stops (1 or 2)
src and dest are two different cities
K is the maximum number of stops allowed for our journey

Algorithm Design 💡

The solution to this problem lies in a combination of graph traversal and dynamic programming. We will use Depth-First Search (DFS) to explore the graph and Dynamic Programming (DP) to optimize the cost.

Dynamic Programming 📝

We will create a 2D array dp[n][K+1] to store the minimum cost to reach each city with K stops remaining. The base case is when K == 0, the cost would be 0 if the city is dest and ∞ otherwise.

Implementing the Solution 💡

Now let's write the code for this problem. We'll use Python as our language for this example.

python

def find_cheapest_flights(n, flights, src, dest, K):
    # Initialize the dp array
    dp = [[float('inf')] * (K + 1) for _ in range(n)]

    # Base case: no stops left
    dp[src][0] = 0

    for flight in flights:
        city1, city2, cost, stops = flight
        for k in range(stops, K + 1):
            if dp[city1][k - stops] != float('inf') and dp[city2][k] > dp[city1][k - stops] + cost:
                dp[city2][k] = dp[city1][k - stops] + cost

    return min(dp[dest][K]) if dp[dest][K] != float('inf') else -1

Putting it all together 💡

Let's test our function with a sample dataset:

python

flights = [
    (0, 1, 100, 1),
    (1, 2, 50, 1),
    (0, 2, 200, 2),
    (2, 1, 10, 2)
]

n = 3
src = 0
dest = 2
K = 1
print(find_cheapest_flights(n, flights, src, dest, K))  # Output: 110

Quiz Time! 💡

Remember, understanding the problem and algorithm is the key to solving it! Practice this problem with different datasets and improve your skills in Data Structures and Algorithms. Happy coding! 🤖