Suppose we are sorting an array of eight integers using quicksort, and we have just finished the first partitioning with the array looking like this: 2 5 1 7 9 12 11 10 Which statement is correct?a)The pivot could be either the 7 or the 9.b)The pivot could be the 7, but it is not the 9c)The pivot is not the 7, but it could be the 9d)Neither the 7 nor the 9 is the pivot.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

Quicksort is a divide-and-conquer algorithm that works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The sub-arrays are then sorted recursively. In the first partitioning step, the pivot can be selected in different ways, but it must be an element of the array.

In the given array of eight integers, the first partitioning step has been performed, and the array looks like this: 2 5 1 7 9 12 11 10. The partitioning step has selected a pivot element and partitioned the other elements into two sub-arrays. To determine which statement is correct, we need to examine the array and the pivot element.

Possible pivot elements:

- The pivot could be either the 7 or the 9: This statement is correct because the pivot must be an element of the array, and both 7 and 9 are elements of the array. Moreover, if the pivot is either 7 or 9, the partitioning would have split the array into two sub-arrays: {2,5,1,7} and {12,11,10,9} or {2,5,1,9} and {12,11,10,7}, depending on the choice of the pivot.
- The pivot could be the 7, but it is not the 9: This statement is incorrect because the pivot could be either 7 or 9, as explained above. Moreover, if the pivot is 7, the partitioning would have split the array into {2,5,1} and {7,9,12,11,10}, which is not a valid partitioning because the sub-array {7,9,12,11,10} contains elements greater than the pivot 7.
- The pivot is not the 7, but it could be the 9: This statement is incorrect because the pivot could be either 7 or 9, as explained above. Moreover, if the pivot is 9, the partitioning would have split the array into {2,5,1,7} and {9,12,11,10}, which is a valid partitioning because all elements in the first sub-array are less than the pivot 9 and all elements in the second sub-array are greater than or equal to the pivot 9.
- Neither the 7 nor the 9 is the pivot: This statement is incorrect because the pivot must be an element of the array, as explained above. Moreover, the partitioning step has already selected a pivot, so it cannot be neither 7 nor 9.

Conclusion:

The correct statement is that the pivot could be either the 7 or the 9. The choice of the pivot affects the partitioning and the subsequent recursive sorting steps of the quicksort algorithm.

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