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

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

Options:

Correct Answer: O(E log V)

Solution:

The time complexity of Dijkstra\'s algorithm using a binary heap 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 using a binary heap?

What is the time complexity of Dijkstra's algorithm 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: Recognize 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: Note that each time we extract the minimum vertex from the heap, it takes O(log V) time because we need to maintain the heap property.
Step 5: Realize that we need to process each edge in the graph. For each edge, we may need to update the distance to a vertex, which also takes O(log V) time.
Step 6: Since there are E edges, and for each edge, we perform an operation that takes O(log V) time, the total time for processing all edges is O(E log V).
Step 7: Combine the time taken for extracting the minimum vertex and processing all edges to conclude that the overall time complexity of Dijkstra's algorithm using a binary heap is O(E log V).

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