Design and Analysis Quick Sort


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It is used on the principle of divide-and-conquer. Quick sort is an algorithm of choice in many situations as it is not difficult to implement. It is a good general purpose sort and it consumes relatively fewer resources during execution.

Advantages

  • It is in-place since it uses only a small auxiliary stack.

  • It requires only n (log n) time to sort n items.

  • It has an extremely short inner loop.

  • This algorithm has been subjected to a thorough mathematical analysis, a very precise statement can be made about performance issues.

Disadvantages

  • It is recursive. Especially, if recursion is not available, the implementation is extremely complicated.

  • It requires quadratic (i.e., n2) time in the worst-case.

  • It is fragile, i.e. a simple mistake in the implementation can go unnoticed and cause it to perform badly.

Quick sort works by partitioning a given array A[p ... r] into two non-empty sub array A[p ... q] and A[q+1 ... r] such that every key in A[p ... q] is less than or equal to every key in A[q+1 ... r].

Then, the two sub-arrays are sorted by recursive calls to Quick sort. The exact position of the partition depends on the given array and index q is computed as a part of the partitioning procedure.

Algorithm: Quick-Sort (A, p, r) 
if p < r then 
   q Partition (A, p, r) 
   Quick-Sort (A, p, q) 
   Quick-Sort (A, q + r, r) 

Note that to sort the entire array, the initial call should be Quick-Sort (A, 1, length[A])

As a first step, Quick Sort chooses one of the items in the array to be sorted as pivot. Then, the array is partitioned on either side of the pivot. Elements that are less than or equal to pivot will move towards the left, while the elements that are greater than or equal to pivot will move towards the right.

Partitioning the Array

Partitioning procedure rearranges the sub-arrays in-place.

Function: Partition (A, p, r) 
x ← A[p] 
i ← p-1 
j ← r+1 
while TRUE do 
   Repeat j ← j - 1 
   until A[j] ≤ x  
   Repeat i← i+1 
   until A[i] ≥ x  
   if i < j then  
      exchange A[i] ↔ A[j] 
   else  
      return j 

Analysis

The worst case complexity of Quick-Sort algorithm is O(n2). However using this technique, in average cases generally we get the output in O(n log n) time.

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