Explain the divide and conquer paradigm from algorithm design with a suitable example. Write the Quick sort algorithm using a randomized approach and explain its time complexity.

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Suresh Chand · Accepted Answer

In the divide and conquer approach, a problem is divided into smaller problems, then the smaller problems are solved independently, and finally, the solutions of smaller problems are combined into a solution for the large problem. Generally, divide-and-conquer algorithms have three parts − Divide the problem into a number of sub-problems that are smaller instances of the same problem. Conquer the sub-problems by solving them recursively. If they are small enough, solve the sub-problems as base cases. Combine the solutions to the sub-problems into the solution for the original problem. Example: A classic example of Divide and Conquer is Merge Sort. In Merge Sort, we divide the array into two halves, sort the two halves recursively, and then merge the sorted halves. Randomized Quick Sort: Randomized quick sort is called randomized if its behavior depends on input as well as the random value generated by the random number generator. The beauty of the randomized algorithm is that no particular input can produce worst-case behavior. Algorithm: RandQuickSort(A, l, r){ if( l < r ){ m = RandPartition(A, l, r); RandQuickSort(A, l, m-1); RandQuickSort(A, m+1, r); } } RandPartition(A, l, r){ k = random(l, r); swap(A[l], A[k]); return Partition(A, l, r); } Partition(A, l, r){ x = l; y = r; p = A[l]; while(x < y){ do{ x++; }while(A[x] <= p); do{ y--; }while(A[y] >= p); if( x < y ){ swap(A[x], A[y]); } } A[l] = A[y]; A[y] = p; return y; } Time Complexity: For the Wrost case, Time complexity is O(n2) For Best case, Time complexity is O(n log n) For Average case, Time complexity is O(n log n)

Explain the divide and conquer paradigm from algorithm design with a suitable example. Write the Quick sort algorithm using a randomized approach and explain its time complexity.

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Explain the divide and conquer paradigm from algorithm design with a suitable example. Write the Quick sort algorithm using a randomized approach and explain its time complexity.

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