In Prim's algorithm, we use decrease key operation. What is the time complexity of this decrease key operation in case of Prim's Algorithm :a)O(n)b)O(nlogn)c)O(logn)d)O(n2)Correct answer is option 'C'. Can you explain this answer?

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Time Complexity of Decrease Key Operation in Prim's Algorithm

To understand the time complexity of the decrease key operation in Prim's algorithm, let's first understand the algorithm itself.

Prim's Algorithm:
Prim's algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a connected weighted graph. It starts with an initial vertex and grows the MST by adding the minimum weight edge at each step until all vertices are included in the MST.

The decrease key operation is used in Prim's algorithm to update the key (weight) of a vertex in the priority queue. This is done when a shorter path to a vertex is found, and it needs to be updated in the priority queue.

Time Complexity of Decrease Key Operation:
The time complexity of the decrease key operation depends on the data structure used to implement the priority queue.

In Prim's algorithm, a binary heap is commonly used as the data structure for the priority queue. In a binary heap, the decrease key operation takes O(logn) time complexity. This is because the operation involves percolating up the updated key to its correct position in the heap, which takes logarithmic time.

Therefore, the correct answer is option 'C' - O(logn).

It's worth mentioning that other data structures, such as Fibonacci heap, can also be used to implement the priority queue in Prim's algorithm. The Fibonacci heap has a better time complexity of O(1) for decrease key operation. However, it has a higher constant factor and is more complex to implement compared to a binary heap.

In conclusion, the time complexity of the decrease key operation in Prim's algorithm is O(logn) when using a binary heap as the data structure for the priority queue.

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