Added Solution for Flight Discount With proper documentation

[vingoel26]

🎯 Problem Information

#139
Problem Name: Flight Discount
Category: Graphs / Shortest Path (Dijkstra’s Algorithm)
CSES Link: https://cses.fi/problemset/task/1195
Difficulty: Medium

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📝 Description

This PR adds the solution for the Flight Discount problem from the CSES Problem Set.
The goal is to find the minimum-cost route from city 1 (Syrjälä) to city n (Metsälä) while optimally using one discount coupon that halves the price of a single flight.
The solution uses a state-based Dijkstra’s algorithm to efficiently handle both the “used” and “unused” discount states for each node.

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🧩 Solution Approach

• Algorithm Used: Modified Dijkstra’s Algorithm with 2D state tracking
• Time Complexity: O((n + m) * log n)
• Space Complexity: O(n + m)
• Key Insights:
  • Maintain two distance arrays for each city:
    • dist[node][0]: Minimum distance without using the discount
    • dist[node][1]: Minimum distance after using the discount
  • Use a min-heap priority queue for efficient shortest path extraction.
  • Process each edge with and without discount consideration.
  • The final result is dist[n][1], representing the minimum cost after discount usage.

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✅ Checklist

Please ensure your PR meets these requirements:

Code Quality

• Solution follows the required template format
• Code is clean and well-commented
• Variable names are descriptive
• Proper error handling for edge cases

Testing

• Solution passes all CSES test cases
• Tested with custom edge cases
• Handles large input constraints efficiently
• No runtime errors or timeouts

Documentation

• Added solution to appropriate category folder
• File name follows naming convention
• Added brief comment explaining the approach
• Updated relevant documentation if needed

Style Guide

• Uses required headers (#include <bits/stdc++.h>)
• Includes fast I/O optimization
• Uses appropriate data types (long long for large numbers)
• Follows competitive programming best practices

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🏷️ Type of Change

• 🐛 Bug fix (fixes an existing solution)
• ✨ New problem solution
• 📚 Documentation improvement
• 🔧 Code optimization/refactoring
• 🎃 Hacktoberfest contribution

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🧪 Testing Details

The solution was tested on both CSES and local environments.

Test Cases Used:
Input:
3 4
1 2 3
2 3 1
1 3 7
2 1 5

Expected Output:
2

Actual Output:
2

Additional tests:

• Verified for large graphs with up to 10⁵ nodes and 2×10⁵ edges.
• Checked correctness when the discount is best applied mid-route, not at the beginning or end.
• Tested with multiple paths having similar costs.

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📸 Screenshots

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📎 Additional Notes

• Implemented with efficient priority queue and state management for discount usage.
• Carefully handles large weights (1 ≤ c ≤ 10⁹).
• Floor division ensures integer correctness in halved cost computation.

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For Maintainers:

• Code review completed
• Solution verified on CSES judge
• Documentation updated if needed
• Labels applied appropriately