graph - Understanding Time complexity calculation for Dijkstra Algorithm

Question

As per my understanding, I have calculated time complexity of Dijkstra Algorithm as big-O notation using adjacency list given below. It didn't come out as it was supposed to and that led me to understand it step by step.

1. Each vertex can be connected to (V-1) vertices, hence the number of adjacent edges to each vertex is V - 1. Let us say E represents V-1 edges connected to each vertex.
2. Finding & Updating each adjacent vertex's weight in min heap is O(log(V)) + O(1) or O(log(V)).
3. Hence from step1 and step2 above, the time complexity for updating all adjacent vertices of a vertex is E*(logV). or E*logV.
4. Hence time complexity for all V vertices is V * (E*logV) i.e O(VElogV).

But the time complexity for Dijkstra Algorithm is O(ElogV). Why?

Answer 1

Dijkstra's shortest path algorithm is O(ElogV) where:

• V is the number of vertices
• E is the total number of edges

Your analysis is correct, but your symbols have different meanings! You say the algorithm is O(VElogV) where:

• V is the number of vertices
• E is the maximum number of edges attached to a single node.

Let's rename your E to N. So one analysis says O(ElogV) and another says O(VNlogV). Both are correct and in fact E = O(VN). The difference is that ElogV is a tighter estimation.