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Shortest Path in a Graph

Certification: Advanced Level Accuracy: 100% Submissions: 3 Points: 15

Write a Python function that implements the Bellman-Ford algorithm to find the shortest path from a source vertex to all other vertices in a weighted graph. The graph may contain negative weight edges, but not negative weight cycles.

Example 1
    
  
  • Input: Edges = [(0, 1, 5), (0, 2, 4), (1, 3, 3), (2, 1, -6), (2, 3, 5)], Source = 0, Vertices = 4
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  • Output: [0, -2, 4, 1]
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  • Explanation: 
    
      
      
    • Step 1: Initialize distances [0, inf, inf, inf].
      • 
      
    • Step 2: Relax all edges (V-1) times.
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    • Step 3: Final distances [0, -2, 4, 1].
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Example 2
    
  
  • Input: Edges = [(0, 1, 4), (0, 2, 2), (1, 2, 3), (1, 3, 2), (1, 4, 3), (2, 3, 1), (3, 4, 5)], Source = 0, Vertices = 5
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  • Output: [0, 4, 2, 3, 7]
    • 
  
  • Explanation:
    
      
      
    • Step 1: Initialize distances [0, inf, inf, inf, inf].
      • 
      
    • Step 2: Relax all edges (V-1) times.
      • 
      
    • Step 3: Final distances [0, 4, 2, 3, 7].
      • 
    
    
  
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Constraints
    
  
  • 1 ≤ number of vertices ≤ 10^3
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  • 0 ≤ number of edges ≤ 10^4
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  • -10^3 ≤ edge weights ≤ 10^3
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  • Graph may contain negative weight edges but not negative weight cycles
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  • Time Complexity: O(V * E), where V is the number of vertices and E is the number of edges
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  • Space Complexity: O(V)
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Hint
Solution

Solution Hints

    
  
  • Initialize distance to vertices as infinity and distance to source as 0
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  • Relax all edges V-1 times, where V is the number of vertices
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  • For each edge (u, v) with weight w, if distance[u] + w < distance[v], update distance[v]
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  • Check for negative weight cycles by attempting to relax edges one more time
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  • Return the distance array containing shortest distances to all vertices from source
    •

Steps to solve by this approach:

 Step 1: Initialize distances from source to all vertices as infinite, and source to itself as 0.

 Step 2: Relax all edges V-1 times, where V is the number of vertices. 
 Step 3: For each edge (u,v,w), if distance[u] + w < distance[v], update distance[v].

 Step 4: Check for negative weight cycles by attempting one more relaxation.

 Step 5: If any distance can still be updated, a negative cycle exists.

 Step 6: Return the final distance array or indication of negative cycle.

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