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If a graph has V vertices and E edges, what is the worst-case time complexity of

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Question: If a graph has V vertices and E edges, what is the worst-case time complexity of Dijkstra\'s algorithm using an adjacency matrix?

Options:

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

Correct Answer: O(V^2)

Solution: 

When using an adjacency matrix, the worst-case time complexity of Dijkstra\'s algorithm is O(V^2) because it requires checking all vertices for the minimum distance.

If a graph has V vertices and E edges, what is the worst-case time complexity of

Practice Questions

Q1
If a graph has V vertices and E edges, what is the worst-case time complexity of Dijkstra's algorithm using an adjacency matrix?
  1. O(V^2)
  2. O(E log V)
  3. O(V + E)
  4. O(V^3)

Questions & Step-by-Step Solutions

If a graph has V vertices and E edges, what is the worst-case time complexity of Dijkstra's algorithm using an adjacency matrix?
  • Step 1: Understand that Dijkstra's algorithm is used to find the shortest path from a starting vertex to all other vertices in a graph.
  • Step 2: Know that an adjacency matrix is a way to represent a graph where we use a 2D array to show the connections (edges) between vertices.
  • Step 3: Realize that in an adjacency matrix, to find the shortest path, we need to check each vertex to see which one has the smallest distance that hasn't been processed yet.
  • Step 4: Since there are V vertices, we will need to check each vertex to find the minimum distance, which takes O(V) time.
  • Step 5: We repeat this process for each vertex, leading to a total of V iterations, each taking O(V) time.
  • Step 6: Therefore, the total time complexity is O(V) for each of the V vertices, resulting in O(V * V) or O(V^2) overall.
  • Dijkstra's Algorithm – An algorithm for finding the shortest paths between nodes in a graph, particularly useful for weighted graphs.
  • Time Complexity – A computational complexity that describes the amount of time it takes to run an algorithm as a function of the length of the input.
  • Adjacency Matrix – A 2D array used to represent a graph, where the element at row i and column j indicates the presence or weight of an edge between vertices i and j.
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