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

In Dijkstra's algorithm, the decrease key operation is performed every time we find a shorter path to a node. The maximum number of times this operation can be performed is directly related to how many nodes (or vertices) are in the graph.

Breakdown

1. Understanding the Graph: In a graph with $n$ nodes, each node can potentially replace its shortest known distance when a shorter path to that node is discovered.
2. Decrease Key Operations: Each time we explore the edges of a node, we may find a shorter path to a neighboring node and perform a decrease key operation.

Relevant Concepts

• In the worst-case scenario, each node will be updated once for all other nodes.
• Thus, for a graph with $n$ nodes, the maximum number of decrease key operations in Dijkstra's algorithm is proportional to the number of nodes.

Analysis and Detail

• If we consider a fully connected graph (also known as a complete graph) where every node is connected to every other node, each node could potentially have its key decreased due to the paths discovered.
• The worst-case scenario would be that every edge leads to a decrease key operation, resulting in $n-1$ operations for every node.

Verify and Summarize

• Since there are $n$ nodes and each can have its key decreased based on the shortest path to it potentially from every other node, the conclusion holds.
• Thus, we would not exceed this in terms of the number of decrease key operations as we process each edge.

Final Answer

The maximum number of times the decrease key operation performed in Dijkstra’s algorithm will be equal to $n - 1$, where $n$ is the number of nodes in the graph.