The maximum number of times the decrease key operation performed in Dijkstra’s algorithm will be equal to ___________ans.

To determine the maximum number of times the decrease key operation is performed in Dijkstra’s algorithm, we need to analyze how the algorithm works and the structure of the priority queue used in it.

1. ### Break Down the Problem

• Dijkstra’s algorithm is designed to find the shortest path from a source vertex to all other vertices in a weighted graph.
• The primary data structure used is a priority queue (often implemented as a min-heap) to efficiently retrieve and update the minimum distances.

2. ### Relevant Concepts

• The decrease key operation is used when we find a shorter path to a vertex. For each vertex $v$ in the graph, if we discover a shorter path to $v$ through another vertex $u$, we update the priority of $v$ in the priority queue.

3. ### Analysis and Detail

• Let's denote $V$ as the number of vertices in the graph.
• In the worst case, Dijkstra’s algorithm might need to perform a decrease key operation for each edge in the graph. This would happen if each vertex has a direct connection to every other vertex.
• The total number of edges $E$ in a complete graph can be represented as $\frac{V(V-1)}{2}$.
• Since the decrease key operation occurs for each outgoing edge from a vertex, the worst-case scenario occurs when every vertex connects to every other vertex, leading to potentially up to $E$ decrease key operations.

4. ### Verify and Summarize

• Therefore, the maximum number of times the decrease key operation can be performed is proportional to the number of edges, leading to $O(E)$ complexity.
• It's important to note that this is the theoretical worst-case scenario, as the performance may vary based on the specific graph structure.

Final Answer

The maximum number of times the decrease key operation performed in Dijkstra’s algorithm will be equal to $O(E)$, where $E$ is the number of edges in the graph.