Reading on Quicksort

Overview

The quicksort is a recursive approach to sorting, and we begin by outlining the principal recursive step. In this step, we make a guess at the value that should end up in the middle of the array. In particular, given the array a[0], ..., a[N-1] of data, arranged at random, then we might guess that the first data item a[0] often should end up in about the middle of the array when the array is finally ordered. (a[0] is easy to locate, and it is as good a guess at the median value as another.) This suggests the following steps:

1. Rearrange the data in the a array, so that A[0] is moved to its proper position. In other words, move a[0] to a[mid], for an appropriate index mid, and rearrange the other elements so that: a[0], a[1], ..., a[mid-1] <- a[mid] and a[mid] <= a[mid+1], ..., a[N-1].
   
   
2. Repeat this process on the smaller lists a[0], a[1], ..., a[mid-1] and a[mid+1], ..., a[N-1].

A specific example is shown below:

The Partition Procedure

As discussed in the above outline, the motivating idea of a Quicksort gives rise to a partition procedure that involves three steps:

1. Choose an array element to serve as a reference value. Typically, this element is called the pivot. (Usually, this is the first element of the array, so this step is easy and quick.)

2. Rearrange elements within an array, so that elements smaller than the pivot are in the first part of the array, and larger elements at the end. Elements equal to the pivot can be anywhere. The resulting array has the form shown in the following schematic:

   

3. Swap the pivot with the last of the smaller elements. (After this step, all elements before the pivot have <= values and all elements after have >= values. Thus, the pivot will be in its correct location if/when the overall array is sorted.) This final result of the partition procedure has the form shown in the next schematic:

   

Since step 1 involves only a decision to identify the pivot (i.e., as the first array element) and step 3 involves only a swap, the overall efficiency of this algorithm depends on rearranging array elements quickly. In most [all?] implementations, the idea is to proceed with a simple loop that examines array elements only once. In all cases, the loop must seek to

• move large elements to the left, and
• move small elements to the right.

Overall, the loop will need to examine successive array elements and determine whether they belong to the left (small) or right (large) grouping. Thus, as processing proceeds, the procedure will have to keep track of three groups: the collection of small elements, the large elements, and the unprocessed elements (that have not been yet been placed in a group).

Several variations for implementing the partition procedure differ in the locations of the small, large, and unprocessed elements, as shown in the following schematic diagram.

Loop Invariant 1: Small Elements first, Then Large, then Unprocessed

With Loop Invariant 1, the idea is to place small elements at the start of the array, just after a[first] and to place large elements at the end of the array. Elements that have yet to be examined are in the middle. As processing proceeds, the middle section gets successively smaller, until all elements have been put into the correct section.

As indicated in the following schematic, variables first and last identify the indices of the array segment being processed, and variables left and right mark the edges of the known small and large collections.

Code implementing Loop Invariant 1 largely follows one of two approaches.

• Examine unprocessed elements down from the right and up from left—swapping only when a misplaced large and small element is identified.
• Work through unprocessed elements from one end only (e.g., work down from the right).

Implementation A: Examine Elements from both ends of the Unprocessed Segment

This traditional approach works from both ends of the array segment toward the middle, comparing data elements to the first element and rearranging the array as necessary. This idea is encapsulated in the following diagram, which is repeated from above.

To implement this approach, we move left and right toward the middle — maintaining the loop invariant with each step.. The details follow:

1. Compare a[first] to a[last], a[last-1], etc. until an element a[right] is found where a[right] < a[first].
   
   
2. Compare a[first] to a[first+1], a[first+2], etc. until an element a[left] is found where a[left] > a[first].
   
   
3. Swap a[left] and a[right].
   At this point,
   • a[first] < a[right], a[right+1], ..., a[last], and
   • a[first] > a[first+1], ..., a[left]
   
   
   
4. Continue steps A and B, comparing the original first element against the end of the arrays, until all elements of the array have been checked.
   
   
5. Swap a[first] with a[right], to put it in its correct location.

These steps are illustrated in the following diagram:

The following code implements this basic step:

int left=first+1;
int right=last;
int temp;

while (right >= left) {
    // search left to find small array item
    while ((right >= left) && (a[first] <= a[right]))
        right--;
    // search right to find large array item
    while ((right >= left) && (a[first] >= a[left]))
        left++;
    // swap large left item and small right item, if needed
    if (right > left) {
        temp = a[left];       // swap a[left] and a[right]
        a[left] = a[right];
        a[right] = temp;
    }
}
// put a[first] in its place
temp = a[first];              // swap the pivot, a[first], and a[right]
a[first] = a[right];
a[right] = temp;

Implementation B: Examine elements from one end of the Unprocessed Segment Only

In this approach, seen in a number of YouTube videos, processing proceeds downward from the top end of the array toward the pivot. The details follow:

1. Compare a[right] with the pivot.

   • If a[right] >= pivot, decrease right, and continue.
   • If a[right] < pivot, swap a[right] and a[left], increase left, and continue.

Altogether, the collection of large elements is expanded whenever a large element is found, and the collection of smal elements is expanded whenever a small element is found.

Although the swap process can be written out within the partition procedure, this approach sometimes (often?) is written with a call to a swap function. This produces the following code:

  int pivot = a[first];
  int left;
  int right = last;
  
  for (left = first+1; left <= right;) {
    if (a[left] < pivot) {
      left++;
    }
    else {
      swap (&a[left], &a[right]);
      right--;
    }
  }

  swap (&a[first], &a[right]);

Loop Invariant 2 with Small Elements on Left; Large Elements Next

In this approach, the idea is to maintain small elements and then large elements toward the start of the array. The right of the array contains elements that are yet to be examined. As processing proceeds, the right section shrinks until all items are in their appropriate section.

During processing, we compare successive items in the array to a[first].

• left marks the array index after the checked array that are less than or equal to a[first]
• right marks the index of the block of array of items that have not been examined yet.

Examining this picture in more detail, at the start, there are no examined items, so there are no items that we know are <= a[first] and also no items that we know are > a[first], as shown in the following diagram:

This picture suggests the following initialization:

   left = first + 1;
   right = first + 1;

Once processing has begun, we examine the unprocessed element a[right].

• If this element is larger than a[first], the loop invariant (and figure) can be adjusted by expanding the > a[first] section: right++
• If this element is less than or equal to a[first], then we need to move it to the collection of small items. A simple way to do that is to swap a[left] and a[right] and increment both left and right.

Processing continues until all items are processed, as shown in the following diagram:

Based on this outline, the entire code for placing a[first] in its correct position is:

 
   // progress through array, 
   //     moving small elements to a[first+1] .. a[left-1]
   //     and moving large elements to a[left] .. a[right-1]
   while (right <= last)
     {
       if (a[first] < a[right])
          right++;
       else {
          // swap a[left] and a[right]
          temp = a[right];
          a[right] = a[left];
          a[left] = temp;
          left++;
          right++;
       }
     }
    // put a[first] in its place
    temp = a[first];
    a[first] = a[left-1];
    a[left-1] = temp;

Loop Invariant 3 with Unprocessed Elements coming before Small and Large elements

In this approach, the segment of unprocessed elements is located immediately after the pivot, with the segments for small and then large elements following.

Effectively, processing for this invariant can be viewed as the mirror image of that for Loop Invariant 2. In this case, processing begins at the right end of the array (with element a[last]) and proceeds toward the beginning of the array. In this processing,

• when small elements are found, the collection of small elements can be expanded.
• when large elements are found, they must be swapped with small ones at the far end of the small collection (at index right), as a means to expand the large collection

With the similarities between the processing for Loop Invariants 2 and 3, further details and the code for Loop Invariant 3 are left to the reader.

Implementing Quicksort

Regardless of the loop invariant used for a partition function, a pivot is identified and other array elements are moved as needed, so that the pivot can be placed after all samaller (or equal) array elements and after all larger (or equal) elements. This positioning of elements is shown in the following schematic, repeated from earlier in this reading:

As part of processing, the new location of the pivot is known and thus could be returned by a partition function.

Based on this partition function, sorting can proceed by applying the partition procedure to the entire array and then recursively to the first and last parts of the array. The base case of the recursion arises if there are no further elements in an array segment to sort.

This gives rise the the following code, called a quicksort.

int partition (int a[ ], int size, int left, int right) {
  int pivot = a[left];
  int l_spot = left+1;
  int r_spot = right;
  int temp;
  
  while (l_spot <= r_spot) {
    while( (l_spot <= r_spot) && (a[r_spot] >= pivot))
      r_spot--;
    while ((l_spot <= r_spot) && (a[l_spot] <= pivot)) 
      l_spot++;

    // if misplaced small and large values found, swap them
    if (l_spot < r_spot) {
      temp = a[l_spot];
      a[l_spot] = a[r_spot];
      a[r_spot] = temp;
      l_spot++;
      r_spot--;
      }
  }

  // swap a[left] with biggest small value
  temp = a[left];
  a[left] = a[r_spot];
  a[r_spot] = temp;

  return r_spot;
}
void quicksorthelper (int a [ ], int size, int left, int right) {
  if (left > right)
    return;
  int mid = partition (a, size, left, right);
  quicksorthelper (a, size, left, mid-1);
  quicksorthelper (a, size, mid+1, right);
}

void quicksort (int a [ ], int n) {
  quicksorthelper (a, n, 0, n-1);
}

Analysis and Timing

The quicksort is called a divide-and-conquer algorithm, because the first step normally divides the array into two pieces and the approach is applied recursively to each piece.

Suppose this code is applied an array containing n randomly ordered data. For the most part, we might expect that the quicksort's divide-and-conquer strategy will divide the array into two pieces, each of size n/2, after the first main step. Applying the algorithm to each half, in turn, will divide the array further -- roughly 4 pieces, each of size n/4. Continuing a third time, we would expect to get about 8 pieces, each of size n/8.

Applying this process i times, would would expect to get about 2ⁱ pieces, each of size n/2ⁱ.

This process continues until each array piece just has 1 element in it, so 1 = n/2ⁱ or 2ⁱ = n or i = log₂ n. Thus, the total number of main steps for this algorithms should be about log₂ n. For each main step, we must examine the various array elements to move the relevant first items of each array segment into their correct places, and this requires us to examine roughly n items.

Altogether, for random data, this suggests that the quicksort requires about log₂ n main steps with n operations per main step. Combining these results, quicksort on random data has O(n log₂ n).

An Improved Quicksort

A basic quicksort algorithm typically chooses the first or last element in an array segment as the point of reference for subsequent comparisons. (This array element is often called the pivot.) This choice works well when the array contains random data, and the work for this algorithm has O(n log n) with random data.

However, when the data in the array initially are in ascending order or descending order, the analysis of efficiency breaks down, and the quicksort has O(n²). (The analysis is much the same as insertion sort for data in descending order.)

To resolve this difficulty, it is common to select an element in the array segment at random to be the pivot. The first part of the quicksort algorithm then follows the following outline:

private static void quicksortKernet (int[] a, int first, int last)
{
   pick an array element a[first], ..., a[last] at random
   swap the selected array element with a[first]

   continue with the quicksort algorithm as described earlier
   ...
}

With this adjustment, quicksort typically performs equally well (O(n log n)) for all types of data.

A Hybrid Quicksort

For the most part, the divide-and-conquer approach of quicksort can be very efficient. Large segments of an array are repeatedly divided in half. This allows the algorithm to utilize roughly O(log₂ n) levels of processing, and the quicksort (usually) runs in time O(n log₂n). However, near the bottom of the recursion, calls to partition can be made to sort array segments with only a few data elements. In such circumstances, the overhead of calling a function may dominate any gain from array processing.

To address such matters, a hybrid approach can be considered. The idea is:

• in the recursive step
  • if the number of elements in an array segment is small
    • use an insertion sort
    • else use partition and the usual quicksort.

Since large array segments are divided in half during processing, this approach retains the speed of a quicksort, but avoids numerous function calls on small array segments. In practice, some experimentation may be needed to determine when an insertion sort should be called.