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What is the time complexity of Dijkstra's algorithm when using a binary heap?

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Question: What is the time complexity of Dijkstra\'s algorithm when using a binary heap?

Options:

  1. O(V^2)
  2. O(E log V)
  3. O(V log V)
  4. O(E + V)

Correct Answer: O(E log V)

Solution: 

When implemented with a binary heap, the time complexity of Dijkstra\'s algorithm is O(E log V), where E is the number of edges and V is the number of vertices.

What is the time complexity of Dijkstra's algorithm when using a binary heap?

Practice Questions

Q1
What is the time complexity of Dijkstra's algorithm when using a binary heap?
  1. O(V^2)
  2. O(E log V)
  3. O(V log V)
  4. O(E + V)

Questions & Step-by-Step Solutions

What is the time complexity of Dijkstra's algorithm when using a binary heap?
  • Step 1: Understand what Dijkstra's algorithm does. It finds the shortest path from a starting vertex to all other vertices in a graph.
  • Step 2: Identify the components of the graph. There are vertices (V) and edges (E). Vertices are the points, and edges are the connections between them.
  • Step 3: Know that Dijkstra's algorithm uses a priority queue to efficiently get the next vertex with the smallest distance. A binary heap is a common way to implement this priority queue.
  • Step 4: Realize that each time we extract the minimum vertex from the binary heap, it takes O(log V) time because we need to maintain the heap property.
  • Step 5: For each vertex, we may need to update the distances to its neighboring vertices. This involves going through all edges (E) connected to the vertex.
  • Step 6: Since we may process each edge once, the total time spent on edge relaxation is O(E).
  • Step 7: Combine the time for extracting the minimum vertex and the time for edge relaxation. The total time complexity becomes O(E log V).
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