If a graph has 5 vertices and 10 edges, what is the maximum time complexity of D

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Question: If a graph has 5 vertices and 10 edges, what is the maximum time complexity of Dijkstra\'s algorithm using an adjacency matrix?

Options:

Correct Answer: O(5^2)

Solution:

Using an adjacency matrix, the time complexity of Dijkstra\'s algorithm is O(V^2), which in this case is O(5^2) or O(25).

If a graph has 5 vertices and 10 edges, what is the maximum time complexity of Dijkstra's algorithm using an adjacency matrix?

If a graph has 5 vertices and 10 edges, what is the maximum time complexity of Dijkstra's algorithm using an adjacency matrix?

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. Here, we have 5 vertices (V) and 10 edges (E).
Step 3: Know that Dijkstra's algorithm can be implemented using an adjacency matrix. An adjacency matrix is a 2D array where the rows and columns represent vertices.
Step 4: Determine the time complexity of Dijkstra's algorithm when using an adjacency matrix. The time complexity is O(V^2), where V is the number of vertices.
Step 5: Substitute the number of vertices into the time complexity formula. Here, V = 5, so we calculate O(5^2).
Step 6: Calculate 5^2, which equals 25. Therefore, the time complexity is O(25).
Step 7: Conclude that the maximum time complexity of Dijkstra's algorithm using an adjacency matrix for this graph is O(25).

Graph Theory – Understanding the properties of graphs, including vertices and edges.
Dijkstra's Algorithm – A shortest path algorithm that can be implemented using different data structures, affecting its time complexity.
Time Complexity – Analyzing the efficiency of algorithms based on the number of vertices and edges.
Adjacency Matrix – A way to represent a graph using a 2D array, which impacts the performance of graph algorithms.