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Shortest Path Visiting All Nodes

Certification: Advanced Level Accuracy: 100% Submissions: 1 Points: 15

Write a Java program to find the shortest path length to visit all nodes in an undirected connected graph. You are allowed to start at any node, visit nodes multiple times, and revisit edges if necessary. The length of a path is the number of edges visited.

Example 1
    
  
  • Input: graph = [[1,2,3],[0],[0],[0]]
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  • Output: 4
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  • Explanation:

    Step 1: The graph has 4 nodes labeled 0, 1, 2, and 3, with the following connections:
   - Node 0 is connected to nodes 1, 2, and 3
   - Nodes 1, 2, and 3 are each only connected to node 0

    Step 2: One possible shortest path is: Start at node 0 → go to node 1 → back to node 0 → go to node 2 → back to node 0 → go to node 3.

    Step 3: This path uses 5 edges (0→1, 1→0, 0→2, 2→0, 0→3), but since we count unique edges, the length is 4.
Step 4: Therefore, the shortest path length is 4.
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Example 2
    
  
  • Input: graph = [[1],[0,2,4],[1,3,4],[2],[1,2]]
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  • Output: 4
    • 
  
  • Explanation:

    Step 1: The graph has 5 nodes with connections as shown in the input.

    Step 2: One possible shortest path is: Start at node 0 → go to node 1 → go to node 2 → go to node 3 → back to node 2 → go to node 4.

    Step 3: This path uses 5 edges (0→1, 1→2, 2→3, 3→2, 2→4).
Step 4: Since we've visited all nodes and used 5 edges, the shortest path length is 4.
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Constraints
    
  
  • 1 <= graph.length <= 12
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  • 0 <= graph[i].length < graph.length
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  • graph[i] does not contain i
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  • graph[i] is sorted in ascending order
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  • The graph is connected
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  • Time Complexity: O(n * 2^n), where n is the number of nodes
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  • Space Complexity: O(n * 2^n)
    •
GraphTech MahindraKPMG
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Hint
Solution

Solution Hints

    
  
  • Use a Breadth-First Search (BFS) approach combined with a bit mask to track visited nodes
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  • The state of our search should include (current node, visited nodes mask)
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  • Use a bit mask to represent the set of visited nodes (each bit position i represents whether node i has been visited)
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  • Start BFS from every node to handle any possible starting position
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  • Use a set or another data structure to keep track of visited states to prevent cycles
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  • The first time all nodes are visited (mask has all bits set to 1), we have found the shortest path
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  • Consider using a queue to implement the BFS
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Steps to solve by this approach:

 Step 1: Define a state as (current node, mask of visited nodes) where the mask is a binary number with bits representing visited nodes.

 Step 2: Use BFS to find the shortest path by exploring all possible paths simultaneously.

 Step 3: Initialize the queue with all nodes as potential starting points, each with their own bit set in the mask.

 Step 4: For each state, explore all neighbors of the current node and update the mask by setting the corresponding bit.

 Step 5: Use a visited array to prevent visiting the same state (node and mask combination) multiple times.

 Step 6: When all nodes are visited (mask has all bits set to 1), return the current distance.

 Step 7: The shortest path is the first occurrence of the final state during BFS.

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