Given an unsorted array. The array has this property that every element in array is at most k distance from its position in sorted array where k is a positive integer smaller than size of array. Which sorting algorithm can be easily modified for sorting this array and what is the obtainable time complexity?a)Insertion Sort with time complexity O(kn)b)Heap Sort with time complexity O(nLogk)c)Quick Sort with time complexity O(kLogk)d)Merge Sort with time complexity O(kLogk)Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Understanding the Problem
The array has a unique property where each element is at most k positions away from its sorted position. This allows us to use a more efficient sorting algorithm than general-purpose ones.
Why Heap Sort?
Heap Sort can be adapted to efficiently sort this nearly sorted array:
- Heap Structure: The elements in the array can be viewed as a nearly complete binary tree. We can maintain a min-heap (or max-heap) to keep track of the smallest (or largest) elements within the k distance.
- K-distance Handling: As we iterate through the array, we can insert elements into the heap, ensuring that we only maintain a heap of size k. This allows us to efficiently find the minimum element in O(1) time.
Time Complexity Breakdown
- Building the Heap: Inserting n elements into a heap of size k takes O(k) time for each insertion. Thus, for n elements, the total time complexity is O(n*k).
- Extracting Elements: Once we have our heap populated, we can extract the minimum element k times, which takes O(log k) time for each extraction. Therefore, extracting all n elements results in O(n log k) time complexity.
Final Complexity
The overall time complexity for sorting the array using this modified Heap Sort approach is O(n log k). This is significantly more efficient than the O(n^2) time complexity of a naïve sorting algorithm like Bubble Sort or Insertion Sort applied to a completely unsorted array.
Conclusion
Thus, the best choice for sorting an array with the described property is Heap Sort, yielding a time complexity of O(n log k).

Answer

1) to sort the array firstly create a min-heap with first k+1 elements and a separate array as resultant array.

2) because elements are at most k distance apart from original position so, it is guranteed that the smallest element will be in this K+1 elements.

3) remove the smallest element from the min-heap(extract min) and put it in the result array.

4) Now,insert another element from the unsorted array into the mean-heap, now,the second smallest element will be in this, perform extract min and continue this process until no more elements are in the unsorted array.finally, use simple heap sort for the remaining elements Time Complexity ------------------------ 1) O(k) to build the initial min-heap 2) O((n-k)logk) for remaining elements... 3) 0(1) for extract min so overall O(k) + O((n-k)logk) + 0(1) = O(nlogk)

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