What is the time complexity of Dijkstra's algorithm using a priority queue imple
Practice Questions
Q1
What is the time complexity of Dijkstra's algorithm using a priority queue implemented with a binary heap?
O(V^2)
O(E log V)
O(V log V)
O(E + V)
Questions & Step-by-Step Solutions
What is the time complexity of Dijkstra's algorithm using a priority queue implemented with 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. A graph consists of 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.
Step 4: Know that a binary heap is a common way to implement a priority queue. It allows for efficient insertion and extraction of the minimum element.
Step 5: Determine how many times Dijkstra's algorithm processes each vertex. It processes each vertex once, which is O(V).
Step 6: Understand that for each vertex processed, the algorithm examines all its edges. The total number of edges examined is O(E).
Step 7: Each time an edge is examined, the algorithm may need to update the priority queue. This operation takes O(log V) time in a binary heap.
Step 8: Combine the steps: For each of the E edges, the algorithm performs an O(log V) operation. Therefore, the total time complexity is O(E log V).
