Flight routes graph algorithms solution added

[AnjaliPai16]

🎯 Problem Information

Problem Name: Flight Routes
Category: graph algorithms
CSES Link: https://cses.fi/problemset/task/1196/
Difficulty: [Medium]

📝 Description

This PR adds an optimized solution for the CSES Flight Routes problem.
The task is to compute the k shortest flight routes from city 1 Syrjälä to city n Metsälä in a directed weighted graph.
A route may visit the same city multiple times, and if multiple paths have the same total cost, they must each be considered separately.
The problem is a variation of Dijkstra’s algorithm, but we must track up to k distinct shortest paths per node.

🧩 Solution Approach

• Algorithm Used: Modified Dijkstra’s Algorithm with Multi-Path Tracking
• Core Idea:

1. Maintain a min-heap (priority queue) to process nodes by their current shortest distance.
2. Each node stores up to k shortest distances in a vector.
3. When a new distance to a node is found:
4. If that node has fewer than k distances stored, we push it.
5. If it already has k distances and the new one is smaller than the largest (worst) stored distance, we replace it.
6. Stop once we have collected k distances for the destination node (n).

Key Optimizations:

1. Use vector for distance lists to handle large weights.
2. Use priority_queue<pair<long long, int>, vector<...>, greater<...>> for min-heap.
3. Skip processing when dist[node].size() == k and the new distance is larger than the worst stored one.
4. Use fast I/O (ios::sync_with_stdio(false); cin.tie(NULL);).

• Time Complexity: O(m * log(n * k))
• Space Complexity: O(n * k)
• Key Insights:

1. Instead of tracking only the single best distance per node (as in standard Dijkstra), we maintain up to k shortest distances for each node.
2. Each node keeps a max-heap of its shortest distances — ensuring insertion and pruning take only O(log k) time.
3. A global min-heap (priority_queue) drives exploration in increasing order of total path cost.
   4)nSince k ≤ 10, this approach remains efficient even for n = 1e5 and m = 2e5.
4. Using a max-heap per node avoids expensive sorting operations and fixes TLE issues.

✅ 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 (snake_case)
• Added brief comment explaining the approach
• Updated any 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

🏷️ Type of Change

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

🧪 Testing Details

Describe how you tested your solution:

1. Ran the solution locally on CSES official input sets.
2. Verified output correctness and performance.
3. Cross-checked with edge cases for repeated nodes and cycles.

Test Cases Used:

Sample Test Case 1:

Input:
4 6 3
1 2 1
1 3 3
2 3 2
2 4 6
3 2 8
3 4 1

Expected Output:
4 4 7
Actual Output:
4 4 7

Custom Test Case 2:

Input:
3 4 2
1 2 2
1 3 5
2 3 2
2 1 1

Expected Output:
4 5
Actual Output:
4 5

📸 Screenshots (if applicable)

📎 Additional Notes

Any additional information, considerations, or context for reviewers:

1. The solution avoids sorting vectors in every iteration — drastically reducing time complexity.
2. Uses a max-heap per node for fast pruning (O(log k) per update).
3. This version is more stable and efficient than the naive vector-based version that caused TLE.

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For Maintainers:

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

[AnjaliPai16]

@ks-iitjmu PR for issue #138 kindly review