Quicksort

Although the shell sort algorithm is significantly better than insertion sort, there is still room for

improvement. One of the most popular sorting algorithms is quicksort. Quicksort executes in

O(n lg n) on average, and O(n2) in the worst-case. However, with proper precautions, worst-case

behavior is very unlikely. Quicksort is a non-stable sort. It is not an in-place sort as stack space

is required. For further reading, consult Cormen [1990].

Theory

The quicksort algorithm works by partitioning the array to be sorted, then recursively sorting

each partition. In Partition (Figure 2-3), one of the array elements is selected as a pivot value.

Values smaller than the pivot value are placed to the left of the pivot, while larger values are

placed to the right.

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Figure 2-3: Quicksort Algorithm

In Figure 2-4(a), the pivot selected is 3. Indices are run starting at both ends of the array.

One index starts on the left and selects an element that is larger than the pivot, while another

index starts on the right and selects an element that is smaller than the pivot. In this case,

numbers 4 and 1 are selected. These elements are then exchanged, as is shown in Figure 2-4(b).

This process repeats until all elements to the left of the pivot are £ the pivot, and all items to the

right of the pivot are ³ the pivot. QuickSort recursively sorts the two sub-arrays, resulting in the

array shown in Figure 2-4(c).

4 2 3 5 1

1 2 3 5 4

1 2 3 4 5

_D_

_E_

_F_

Lb Ub

SLYRW

Lb M Lb

Figure 2-4: Quicksort Example

As the process proceeds, it may be necessary to move the pivot so that correct ordering is

maintained. In this manner, QuickSort succeeds in sorting the array. If we’re lucky the pivot

selected will be the median of all values, equally dividing the array. For a moment, let’s assume

int function Partition (Array A, int Lb, int Ub);

begin

select a pivot from A[Lb]…A[Ub];

reorder A[Lb]…A[Ub] such that:

all values to the left of the pivot are £ pivot

all values to the right of the pivot are ³ pivot

return pivot position;

end;

procedure QuickSort (Array A, int Lb, int Ub);

begin

if Lb < Ub then

M = Partition (A, Lb, Ub);

QuickSort (A, Lb, M – 1);

QuickSort (A, M + 1, Ub);

end;

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that this is the case. Since the array is split in half at each step, and Partition must eventually

examine all n elements, the run time is O(n lg n).

To find a pivot value, Partition could simply select the first element (A[Lb]). All other

values would be compared to the pivot value, and placed either to the left or right of the pivot as

appropriate. However, there is one case that fails miserably. Suppose the array was originally in

order. Partition would always select the lowest value as a pivot and split the array with one

element in the left partition, and Ub – Lb elements in the other. Each recursive call to quicksort

would only diminish the size of the array to be sorted by one. Therefore n recursive calls would

be required to do the sort, resulting in a O(n2) run time. One solution to this problem is to

randomly select an item as a pivot. This would make it extremely unlikely that worst-case

behavior would occur.

Implementation

The source for the quicksort algorithm may be found in file qui.c. Typedef T and comparison

operator compGT should be altered to reflect the data stored in the array. Several enhancements

have been made to the basic quicksort algorithm:

· The center element is selected as a pivot in partition. If the list is partially ordered,

this will be a good choice. Worst-case behavior occurs when the center element happens

to be the largest or smallest element each time partition is invoked.

· For short arrays, insertSort is called. Due to recursion and other overhead, quicksort

is not an efficient algorithm to use on small arrays. Consequently, any array with fewer

than 12 elements is sorted using an insertion sort. The optimal cutoff value is not critical

and varies based on the quality of generated code.

· Tail recursion occurs when the last statement in a function is a call to the function itself.

Tail recursion may be replaced by iteration, resulting in a better utilization of stack space.

This has been done with the second call to QuickSort in Figure 2-3.

· After an array is partitioned, the smallest partition is sorted first. This results in a better

utilization of stack space, as short partitions are quickly sorted and dispensed with.

Included in file qsort.c is the source for qsort, an ANSI-C standard library function usually

implemented with quicksort. Recursive calls were replaced by explicit stack operations. Table

2-1 shows timing statistics and stack utilization before and after the enhancements were applied.

count before after before after

16 103 51 540 28

256 1,630 911 912 112

4,096 34,183 20,016 1,908 168

65,536 658,003 470,737 2,436 252

time ( m s) stacksize

Table 2-1: Effect of Enhancements on Speed and Stack Utilization