Solution for Message Routes

[vingoel26]

🎯 Problem Information

Problem Name: Message Route
Category: Graph (Shortest Path, BFS)
CSES Link: https://cses.fi/problemset/task/1667
Difficulty: [Medium]
#111

📝 Description

This PR adds a clean, well-documented C++ solution for the Message Route problem from CSES.
The program determines whether a message can be sent from computer 1 to computer n in an undirected network, and if so prints the shortest route (in terms of number of computers visited). If there is no route, it prints IMPOSSIBLE.

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🧩 Solution Approach

• Algorithm Used: Breadth-First Search (BFS) on an unweighted undirected graph.
• Time Complexity: O(N + M) — each vertex and edge is processed at most once.
• Space Complexity: O(N + M) — adjacency list + distance + parent arrays.
• Key Insights:
  • BFS from node 1 finds shortest distances (in edges) to every reachable node because it explores level-by-level.
  • Track parent[node] while doing BFS to reconstruct the shortest path from 1 to n after traversal.
  • If distance[n] remains INF, node n is unreachable → output IMPOSSIBLE.
  • Path length reported should be number of nodes on the route = distance[n] + 1.

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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 (inline comments explain BFS, distance and parent usage)
• Variable names are descriptive (adj, distance, parent, q)
• Proper error/edge-case handling (IMPOSSIBLE when disconnected, handles n = 1)

Testing

• Solution passes all CSES test cases
• Tested with custom edge cases (disconnected graphs, single node, multiple shortest paths)
• Handles large input constraints efficiently (N ≤ 1e5, M ≤ 2e5)
• No runtime errors or timeouts (uses iterative BFS; no recursion)

Documentation

• Added solution to appropriate category folder (graphs/)
• File name follows naming convention (message_route.cpp)
• Added brief comment explaining the approach (top of file header block)
• Updated any relevant documentation if needed

Style Guide

• Uses required header (#include <bits/stdc++.h>)
• Includes fast I/O optimization (ios::sync_with_stdio(false); cin.tie(nullptr);)
• Uses appropriate data types (int acceptable here; long long not needed)
• Follows competitive programming best practices (1-indexed adjacency, INF sentinel)

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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

• Verified correctness on CSES judge (Accepted).
• Ran multiple custom tests including edge and stress cases (random graphs up to limits).
• Verified no stack overflow (BFS uses queue), and memory usage is within bounds.

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

Expected Output:
4
1 2 4 5

Actual Output:
4
1 2 4 5

Input:
4 2
1 2
3 4

Expected Output:
IMPOSSIBLE

Actual Output:
IMPOSSIBLE

Input:
1 0

Expected Output:
1
1

Actual Output:
1
1

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📸 Screenshots (if applicable)

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📎 Additional Notes

• The implementation is iterative (BFS) and avoids recursion limits.
• Reconstructing the path uses parent[] and reverse() to output 1 → ... → n.
• The code prints the number of nodes on the route followed by the route nodes separated by spaces.

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For Maintainers:

• Code review completed
• Solution verified on CSES judge
• Documentation updated if needed
• Labels applied appropriately (suggested: graphs, good first issue, hacktoberfest)