What is the time complexity of Dijkstra's algorithm using a priority queue?
Practice Questions
Q1
What is the time complexity of Dijkstra's algorithm using a priority queue?
O(V^2)
O(E + V log V)
O(V log V)
O(E log V)
Questions & Step-by-Step Solutions
What is the time complexity of Dijkstra's algorithm using a priority queue?
Step 1: Understand what Dijkstra's algorithm does. It finds the shortest path from a starting node to all other nodes in a graph.
Step 2: Identify the components of the graph. A graph has vertices (nodes) and edges (connections between nodes). Let V be the number of vertices and E be the number of edges.
Step 3: Learn about the priority queue. A priority queue helps efficiently get the next node with the smallest distance during the algorithm's execution.
Step 4: Analyze the main operations in Dijkstra's algorithm. The algorithm repeatedly selects the node with the smallest distance and updates the distances of its neighbors.
Step 5: Count how many times we need to perform operations. We will process each vertex (V times) and for each vertex, we may need to update the distances of its edges (E times).
Step 6: Understand the cost of using a priority queue. Each time we insert or extract the minimum element from the priority queue, it takes O(log V) time.
Step 7: Combine the operations. For each of the E edges, we may perform an operation that takes O(log V) time, leading to a total time of O(E log V).
Step 8: Conclude that the overall time complexity of Dijkstra's algorithm using a priority queue is O(E log V).
