Question

In the proof of correctness of Dijkstra's algorithm, we use induction on k. What does k represent?

a) The number of explored nodes
b) The length of the shortest path
c) The number of edges considered
d) The number of active nodes

Answer 1

In the proof of correctness for Dijkstra's algorithm, k represents the number of explored nodes, and induction on k is used to demonstrate the algorithm's accuracy in determining the shortest paths.

Step-by-step explanation:

In the proof of correctness for Dijkstra's algorithm, induction on k is used to show that the algorithm correctly finds the shortest path from the starting node to every other node in the graph. Here, k typically represents the number of explored nodes. This means k is the number of nodes for which the algorithm has determined the shortest path up to the current point in its execution. The proof uses induction to show that if the algorithm works correctly for finding the shortest path to k nodes, when we add one more node to make it k+1, the algorithm will still yield the correct shortest path for this extended set of nodes.