A Survey of Adaptive QuickSort Algorithms

Laila Khreisat
International Journal of Computer Science and Security (IJCSS), Volume (12) : Issue (1) : 2018 1
A Survey of Adaptive QuickSort Algorithms
Laila Khreisat khreisat@fdu.edu
Dept. of Math, Computer Science, and Physics
Fairleigh Dickinson University
285 Madison Ave, Madison NJ 07940 USA
Abstract
In this paper, a survey of adaptive quicksort algorithms is presented. Adaptive quicksort
algorithms improve on the worst case behavior of quicksort when the list of elements is sorted or
nearly sorted. These algorithms take into consideration the already existing order in the input list
to be sorted. A detailed description of each algorithm is provided. The paper provides an
empirical study of these algorithms by comparing each algorithm in terms of the number of
comparisons performed and the running times when used for sorting arrays of integers that are
already sorted, sorted in reverse order, and generated randomly.
Keywords: Algorithm, Survey, Quicksort, Sorting, Adaptive Sorting.
1. INTRODUCTION
Since its development by [1], the quicksort algorithm has been considered the most popular and
most efficient internal sorting algorithm. It has been widely studied and described [2, 3, 4, 5, 6, 7].
A divide-and-conquer algorithm, Quicksort sorts an array S of n elements by partitioning the array
into two parts, placing small elements on the left and large elements on the right, and then
recursively sorting the two subarrays. The major drawback of the Quicksort algorithm is that when
the array to be sorted is already sorted or is partially sorted, the algorithm will require
)( 2
nO comparisons. To improve on the worst-case behavior of Quicksort, several adaptive
sorting algorithms have been developed. These algorithms take into consideration the already
existing order in the input list, see [8]. Insertion sort is an adaptive sorting algorithm. Wainwright
[9] developed Bsort, an adaptive sorting algorithm, which improves the average behavior of
Quicksort and eliminates the worst-case behavior for sorted or nearly sorted lists. Qsorte, is
another adaptive algorithm also developed by Wainright [10], which performs as well as Quicksort
for lists of random values, and performs O(n) comparisons for sorted or nearly sorted lists.
Another adaptive sorting algorithm is Nico, which was developed by [11]. It is a version of
Quicksort in which the partition function uses a for loop to roll the largest keys in the array to the
bottom of the array.
Introsort (introspective sort) [12] is a hybrid sorting algorithm that uses both quicksort and
heapsort. The method starts with quicksort and when the recursion depth goes beyond a
specified threshold it switches to heapsort. Thus the algorithm combines the good aspects of both
algorithms.
The paper studies these four sorting algorithms by comparing each algorithm in terms of the
number of comparisons performed and the running times when used for sorting arrays of integers
that are already sorted, sorted in reverse order, and generated randomly. The algorithms are also
compared against the original quicksort algorithm, which we call Hoare.
2. SORTING ALGORITHMS
The original Quicksort algorithm developed by [1] is considered to be the most popular and most
efficient internal sorting algorithm. It has been widely studied and described [2, 3, 4, 5, 6, 7]. The

Laila Khreisat
algorithm’s worst-case time complexity of )( 2
nO occurs when the list of elements is already
sorted or nearly sorted, or sorted in reverse order, [9]. The efficiency of Quicksort ultimately
depends on the choice of the pivot see [13]. The choice of the pivot produces different variations
of the algorithm. A pivot value that divides the list of keys near the middle is considered the ideal
choice. In Hoare’s original algorithm the pivot was chosen at random and the choice resulted in
1.386nlgn expected comparisons see [14].
The following is the pseudo-code for the Quicksort algorithm:
void quicksort(int A[ ], int L, int R)
//Sorting starts by calling partition, which will choose a pivot and place the pivot A[i] in
//its correct position. Then the recursive calls to quicksort will rearrange the elements in
//the array around the pivot such that the array A will be sorted.
{
int i;
if (R <= L) return;
i = partition(A, L, R);
quicksort(A, L, i-1);
quicksort(A, i+1, R);
}
void partition(int first,int last, int& pos)
{
int p,l,r;
l = first;
r = last;
p = l;
swap(l,rand()%(last-first+1)+first);
while (l < r)
{
while ((l < r)&& (ar[p] <= ar[r]))
{ r--; }
swap(p,r);
p = r;
while ((l < r)&&(ar[p] >= ar[l]))
{ l++; }
swap(p,l);
p = l;
}
pos = p;
}
Another variation of the Quicksort algorithm was developed by [11] based on an algorithm
suggested by Nico Lomuto. In this algorithm the partition function uses a for loop to roll the
largest keys in the array to the bottom of the array. A pivot T is chosen at random, and an index
called LastLow is computed and used to rearrange the array X such that all keys less than the
pivot T are on one side of the index LastLow, while all other keys are on the other side of the
index. This achieved using a for loop that scans the array from left to right, using the variables I
and LastLow as indices to maintain the following invariant in array X:

Laila Khreisat
< T >= T ?
A LastLow I B
If X[I] >= T then the invariant is still valid . If X[I]<T, the invariant is regained by incrementing
LastLow by 1 and then swapping X[I] and X[Lastlow]. The following is a C++ implementation of
the partition function:
void Partition(int low, int high, int& pos)
{
int i, j;
int pivot;
swap(low,rand()%(high-low+1)+low);
pivot = ar[low];
i = low;
for (j = low+1; j <= high; j++) //j = I
{
if (ar[j] < pivot)
{
i = i +1; // i = Lastlow
swap(i, j);
}
}
swap(low, i);
pos = i;
}
Wainwright [9] developed Bsort, a variation of Quicksort, designed for nearly sorted lists and lists
that are nearly sorted in reverse order. The author claimed that the algorithm performs
)log( 2 nnO comparisons for all distribution of keys. However, the claim was disproved in a
technical correspondence in 1986 [15] that showed that the algorithm exhibits )( 2
nO behavior.
For lists that are sorted or sorted in reverse order, the algorithm performs )(nO comparisons.
Bsort uses the interchange technique from Bubble sort in combination with the traditional
Quicksort algorithm. During each pass of the algorithm the middle key is chosen as the pivot and
then the algorithm switches over to Quicksort. Each key that is placed in the left subarray will be
placed at the right end of the subarray. If the key is not the first key in the subarray, it will be
compared with its left neighbor to make sure that the pair of keys is in sorted order. If the new key
does not preserve the order of the subarray, it will be swapped with its left neighbor. Similarly,
each new key that is placed in the right subarray, will be placed at the left end of the subarray and
if it is not the first key, it will be compared with its right neighbor to make sure that the pair of keys
is in sorted order, if not the two keys will be swapped [16]. This process ensures that the
rightmost key in the left subarray will be the largest value, and the leftmost key in the rightmost
subarray will be the smallest value, at any point during the execution of the algorithm.
Qsorte is a quicksort algorithm developed by Wainwright [10] that includes the capability of an
early exit for sorted arrays. It is a variation of the original Quicksort algorithm with a modified
partition phase, where the left and right subarrays are checked if they are sorted or not.
During the partitioning phase the middle key is chosen as the pivot. Initially, both the left and right
subarrays are assumed to be sorted. When a new key is placed in the left subarray, and the
subarray is still sorted, then if the subarray is not empty, the new key will be compared with its left

Laila Khreisat
neighbor. If the two keys are not in sorted order then the subarray is marked as unsorted, and the
keys are not swapped. Similarly, when a new key is placed in the right subarray, and the subarray
is still sorted, then if the subarray is not empty, the new key will be compared with its right
neighbor. If the two keys are not in sorted order then the subarray is marked as unsorted, and the
keys are not swapped. At the end of the partitioning phase, any subarray that is marked as sorted
will not be partitioned, [16]. Qsorte’s worst-case time complexity is )( 2
nO which happens when
the chosen pivot is always the smallest value in the subarray. During this scenario Qsorte will
repeatedly partition the subarray into two subarrays with only one key in one of the subarrays.
void Qsorte (int m, int n)
{
int k, v;
bool lsorted, rsorted;
if ( m < n ){
FindPivot (m, n, v);
Partition (m, n, k, lsorted, rsorted);
if (! lsorted) Qsorte(m, k-1);
if (! Rsorted) Qsorte(k, n);
}
}
Introsort (introspective sort) [12] is a hybrid sorting algorithm that uses both quicksort and
heapsort. The algorithm starts by using quicksort and switches to heapsort when the recursion
depth exceeds a specified threshold, thus avoiding the )( 2
nO worst-case behavior of quicksort.
3. EMPIRICAL TESTING AND RESULTS
To study the performance of the sorting algorithms described in section 2, all algorithms were
used for sorting arrays of integers that were already sorted, sorted in reverse order, and
generated randomly. The experiments were conducted on a computer with an Intel i7 processor
with a speed of 2.6 GHz, and 16 GB of RAM. The sizes of the arrays ranged from N = 3000 to N
= 400,000 elements. For the case of arrays of random numbers, each algorithm was used to sort
three sequences of random numbers of a specific size N, and the average running time and
number of comparisons were calculated. The Mersenne Twister random number generator
developed by Matsumoto and Nishimura [17] was used to generate the sequences of random
numbers. The generator has a period of 1219937
 .
Figure 1 below shows the running times for the sorting algorithms when used to sort arrays of
random numbers of different sizes. The fastest algorithm is Qsorte, giving the best performance
for sorting arrays of random integers overall. This is followed by Bsort. Hoare is faster than Nico
for data sizes between N= 9000 and N=262144, after which the two algorithms are comparable.
The slowest algorithm was Introsort.
In terms of the number of comparisons, Nico and Introsort were comparable and had the best
performance compared to the other algorithms. Qsorte required on average 1.15 more
comparisons than Nico, and Hoare performed on average 1.23 more comparisons than Qsorte.
The worst performance was exhibited by Bsort requiring on average 1.7 more comparisons than
Hoare, and 2.3 more comparisons than Nico. The performance of all the algorithms in terms of
the number of comparisons was of order )log( 2 NNO , see figure 2.

Laila Khreisat
FIGURE 1: Average Running Times for Random Data.
FIGURE 2: Average Number of Comparisons for Random Data.
For data sorted in reverse order, Bsort was the fastest for sorting a list of size 10,000 or less,
followed by Qsorte, then Hoare, with Introsort having the slowest performance, see figure 3-1.
For lists of size 100,000 and above, figure 3-2 clearly shows that Qsorte had the best
performance, while Introsort exhibited the slowest running time.

Laila Khreisat
FIGURE 3-1: Running times for Data Sorted in Reverse.
FIGURE 3-2: Running times for Data Sorted in Reverse.
In terms of the number of comparisons, Qsorte required the smallest number of comparisons
followed by Bsort, both of which were of order )(NO . Comparing Nico and Hoare, it is apparent
that Hoare requires more comparisons than Nico and both are of order )log( 2 NNO . Hoare
made the largest number of comparisons overall, and on average Hoare made 14 more
comparisons than Qsorte, which exhibited the best performance. The behavior of all three
algorithms, namely, Hoare, Nico, and Introsort, in terms of the number of comparisons was of
order )log( 2 NNO , see figures 4-1 and 4-2.

Laila Khreisat
FIGURE 4-1: Number of Comparisons for Data Sorted in Reverse.
FIGURE 4-2: Number of Comparisons for Data Sorted in Reverse.
For sorted data, both Bsort and Qsorte achieved the fastest running times. Bsort and Qsorte
were both faster than Hoare which was faster than Nico, and Introsort exhibited the worst
behavior, see figure 5.
Qsorte and Bsort performed the same number of comparisons, which was of order )(NO , see
figure 6-1. This is the expected behavior of Bsort and Qsorte for sorted lists. Introsort
performed the smallest number of comparisons for N>=200,000 compared to Hoare and Nico. All
three algorithms exhibited )log( 2 NNO behavior, see figure 6-2.

Laila Khreisat
FIGURE 5: Running Times for Sorted Data.
FIGURE 6-1: Number of Comparisons for Sorted Data.

Laila Khreisat
FIGURE 6-2: Number of Comparisons for Sorted Data.
4. CONCLUSIONS
In this paper the results of an empirical study of adaptive variations of the quicksort were
presented. All the sorting algorithms were studied by computing their running times and number
of comparisons made when used for sorting random data, sorted data and data sorted in reverse.
Qsorte was the fastest when used for sorting random data, while Bsort was the fastest for
sorting lists of data already sorted in reverse of size 10,000 or less. For lists of size 100,000 and
above, Qsorte had the best performance. For sorted data, both Bsort and Qsorte achieved the
fastest running times.
In terms of the number of comparisons for random data, Nico and Introsort were comparable
and had the best performance, which was of order )log( 2 NNO . For data sorted in reverse,
Qsorte required the smallest number of comparisons followed by Bsort, both of which were of
order )(NO . For sorted data Qsorte and Bsort performed the same number of comparisons,
which was of order )(NO . Introsort performed the smallest number of comparisons for
N>=200,000 compared to Hoare and Nico. All three algorithms exhibited )log( 2 NNO .
5. REFERENCES
[1] C.A.R. Hoare. “Algorithm 64: Quicksort.” Communications of the ACM, vol. 4, pp. 321, Jul.
1961.
[2] R. Loeser. “Some performance tests of: quicksort: and descendants.” Communications of
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[4] R. Chaudhuri, A. C. Dempster. “A note on slowing Quicksort,” in Proc. SIGCSE , 1993, vol.
25.
[5] R. Sedgewick. “Quicksort.” PhD dissertation, Stanford University, Stanford, CA, USA, 1975.

Laila Khreisat
[6] R. Sedgewick. “The Analysis of Quicksort Programs.” Acta Informatica, vol. 7, pp. 327–355,
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[7] R. Sedgewick. “Implementing Quicksort programs.” Communications of the ACM, vol.
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[8] K. Mehlhorn. “Data Structures and Algorithms,” in EATCS Monographs on Theoretical
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ACM, vol. 28(4), pp. 396-402, April 1985.
[10] R.L. Wainright. “Quicksort algorithms with an early exit for sorted subfiles.” Communications
of the ACM, pp. 183-190, 1987.
[11] J. Bentley. “Programming Pearl: How to sort.” Communications of the ACM, vol. 27(4),
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[12] D.R. Musser. Introspective Sorting and Selection Algorithms. Software: Practice and
Experience. Wiley, Vol. 27(8), 1997, pp. 983–993.
[13] R. Sedgewick. Algorithms in C++. Addison Wesley, 1998.
[14] C.A.R. Hoare. “Quicksort.” Computer Journal, vol. 5, pp. 10–15, 1962.
[15] CORPORATE Tech Correspondence, Technical correspondence. Communications of the
ACM, vol. 29(4), pp. 331-335, Apr. 1986.
[16] L. Khreisat. “QuickSort: A Historical Perspective and Empirical Study.” International Journal
of Computer Science and Network Security, vol. 7(12), Dec. 2007.
[17] M. Matsumoto, T. Nishimura. “Mersenne Twister: A 623-dimensionally equidistributed
uniform pseudorandom number generator.” ACM Trans. on Modeling and Computer
Simulation, vol. 8(1), pp.3-30, Jan. 1998.

A Survey of Adaptive QuickSort Algorithms

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