What is the time complexity of Dijkstra's algorithm when using a binary heap?
Practice Questions
Q1
What is the time complexity of Dijkstra's algorithm when using 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 when 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: Know 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: Realize that each time we extract the minimum vertex from the binary heap, it takes O(log V) time because we need to maintain the heap property.
Step 5: For each vertex, we may need to update the distances to its neighboring vertices. This involves going through all edges (E) connected to the vertex.
Step 6: Since we may process each edge once, the total time spent on edge relaxation is O(E).
Step 7: Combine the time for extracting the minimum vertex and the time for edge relaxation. The total time complexity becomes O(E log V).
