What is the running time of Bellman Ford Algorithm when graph is Complete graph*1 pointO(V2)O(O(V3))O(VE)O(V)

To determine the running time of the Bellman-Ford algorithm in the context of a complete graph, we can break down the problem as follows:

1. ### Break Down the Problem

• Identify the number of vertices (V) and edges (E) in a complete graph.
• Examine the typical operation of the Bellman-Ford algorithm and its time complexity.

2. ### Relevant Concepts

• In a complete graph, every vertex is connected to every other vertex.
• The number of edges in a complete graph with V vertices is given by $E = \frac{V(V - 1)}{2}$.

3. ### Analysis and Detail

• The Bellman-Ford algorithm iterates V - 1 times over all edges to update the shortest paths from the source vertex to other vertices.
• Therefore, the time complexity of the Bellman-Ford algorithm can be expressed as: $O(V \times E)$
• In a complete graph: $E = \frac{V(V - 1)}{2} \implies O(V^2)$

4. ### Verify and Summarize

• Consequently, substituting $E$ in the Bellman-Ford complexity gives us: $O(V \times \frac{V(V - 1)}{2}) = O(V^3)$
• Thus, the corrected interpretation of Bellman-Ford in terms of its operation on a complete graph leads us to the conclusion that the running time is cubic in relation to the number of vertices.

Final Answer

The running time of the Bellman-Ford algorithm on a complete graph is $O(V^3)$.