21. Sorting

Binary searches only work on lists that are in order. So how do programs get a list in order? How does a program sort a list of items when the user clicks a column heading, or otherwise needs something sorted?

There are several algorithms that do this. The two easiest algorithms for sorting are the selection sort and the insertion sort. Other sorting algorithms exist as well, such as the shell, merge, heap, and quick sorts.

The best way to get an idea on how these sorts work is to watch them. To see common sorting algorithms in action visit this excellent website:

http://www.sorting-algorithms.com

Each sort has advantages and disadvantages. Some sort a list quickly if the list is almost in order to begin with. Some sort a list quickly if the list is in a completely random order. Other lists sort fast, but take more memory. Understanding how sorts work is important in selecting the proper sort for your program.

21.1. Swapping Values

Before learning to sort, we need to learn how to swap values between two variables. This is a common operation in many sorting algorithms. Suppose a program has a list that looks like the following:

my_list = [15,57,14,33,72,79,26,56,42,40]

The developer wants to swap positions 0 and 2, which contain the numbers 15 and 14 respectively. See Figure 18.1.

../../_images/sortgraph1.svg

Figure 18.1: Swapping values in an array

A first attempt at writing this code might look something like this:

my_list[0] = my_list[2]
my_list[2] = my_list[0]
../../_images/sortgraph2.svg

Figure 18.2: Incorrect attempt to swap array values

See Figure 18.2 to get an idea on what would happen. This clearly does not work. The first assignment list[0] = list[2] causes the value 15 that exists in position 0 to be overwritten with the 14 in position 2 and irretrievably lost. The next line with list[2] = list[0] just copies the 14 back to cell 2 which already has a 14.

To fix this problem, swapping values in an array should be done in three steps. It is necessary to create a temporary variable to hold a value during the swap operation. See Figure 18.3. The code to do the swap looks like the following:

Swapping two values in an array
temp = my_list[0]
my_list[0] = my_list[2]
my_list[2] = temp

The first line copies the value of position 0 into the temp variable. This allows the code to write over position 0 with the value in position 2 without data being lost. The final line takes the old value of position 0, currently held in the temp variable, and places it in position 2.

../../_images/sortgraph1b.svg

Figure 18.3: Correct method to swap array values

21.2. Selection Sort

The selection by looking at element 0. Then code next scans the rest of the list from element 1 to n-1 to find the smallest number. The smallest number is swapped into element 0. The code then moves on to element 1, then 2, and so forth. Graphically, the sort looks like Figure 18.4.

../../_images/sortgraph.svg

Figure 18.4: Selection Sort

The code for a selection sort involves two nested loops. The outside loop tracks the current position that the code wants to swap the smallest value into. The inside loop starts at the current location and scans to the right in search of the smallest value. When it finds the smallest value, the swap takes place.

Selection Sort
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
def selection_sort(my_list):
    """ Sort a list using the selection sort """

    # Loop through the entire array
    for cur_pos in range(len(my_list)):
        # Find the position that has the smallest number
        # Start with the current position
        min_pos = cur_pos

        # Scan left to right (end of the list)
        for scan_pos in range(cur_pos + 1, len(my_list)):

            # Is this position smallest?
            if my_list[scan_pos] < my_list[min_pos]:

                # It is, mark this position as the smallest
                min_pos = scan_pos

        # Swap the two values
        temp = my_list[min_pos]
        my_list[min_pos] = my_list[cur_pos]
        my_list[cur_pos] = temp

The outside loop will always run \(n\) times. The inside loop will run an average of \(\frac{n}{2}\) times per run of the outside loop. Therefore the inside loop will run a total of \(n \cdot \frac{n}{2}\) or \(\frac{n^2}{2}\) times.

This will be the case regardless if the list is in order or not. The loops’ efficiency may be improved by checking if min_pos and cur_pos are equal before line 20. If those variables are equal, there is no need to do the three lines of swap code.

In order to test the selection sort code above, the following code may be used. The first function will print out the list. The next code will create a list of random numbers, print it, sort it, and then print it again. On line 5 the print statement right-aligns the numbers to make the column of numbers easier to read. Formatting print statements will be covered in a later chapter.

Code to create and print list to sort
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
# Before this code, paste the selection sort and import random

def print_list(my_list):
    for item in my_list:
        print("{:3}".format(item), end="")
    print()

# Create a list of random numbers
my_list = []
for i in range(10):
    my_list.append(random.randrange(100))

# Try out the sort
print_list(my_list)
selection_sort(my_list)
print_list(my_list)

See an animation of the selection sort at:

http://www.sorting-algorithms.com/selection-sort

For a truly unique visualization of the selection sort, search YouTube for “selection sort dance” or use this link:

http://youtu.be/Ns4TPTC8whw

You also can trace through the code using Selection Sort on Python Tutor.

21.3. Insertion Sort

The insertion sort is similar to the selection sort in how the outer loop works. The insertion sort starts at the left side of the array and works to the right side. The difference is that the insertion sort does not select the smallest element and put it into place; the insertion sort selects the next element to the right of what was already sorted. Then it slides up each larger element until it gets to the correct location to insert. Graphically, it looks like Figure 18.5.

../../_images/sortgraph4.svg

Figure 18.5: Insertion Sort

The insertion sort breaks the list into two sections, the “sorted” half and the “unsorted” half. In each round of the outside loop, the algorithm will grab the next unsorted element and insert it into the list.

In the code below, the key_pos marks the boundary between the sorted and unsorted portions of the list. The algorithm scans to the left of key_pos using the variable scan_pos. Note that in the insertion sort, scan_pos goes down to the left, rather than up to the right. Each cell location that is larger than key_value gets moved up (to the right) one location.

When the loop finds a location smaller than key_value, it stops and puts key_value to the left of it.

The outside loop with an insertion sort will run \(n\) times. For each run of the outside loop, the inside loop will run an average of \(\frac{n}{4}\) times if the loop is randomly shuffled. In total, the inside loop would run \(n\cdot\frac{n}{4}\) times, or simplified, \(\frac{n^2}{4}\) times.

What’s really important: If the loop is close to a sorted loop already, then the inside loop does not run very much, and the sort time is closer to n. The insertion sort is the fastest sort for nearly-sorted lists. If the list is reversed, then the insertion sort is terrible.

The selection sort doesn’t really care what order the list is in to begin with. It performs the same regardless.

Insertion Sort
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
def insertion_sort(my_list):
    """ Sort a list using the insertion sort """

    # Start at the second element (pos 1).
    # Use this element to insert into the
    # list.
    for key_pos in range(1, len(my_list)):

        # Get the value of the element to insert
        key_value = my_list[key_pos]

        # Scan from right to the left (start of list)
        scan_pos = key_pos - 1

        # Loop each element, moving them up until
        # we reach the position the
        while (scan_pos >= 0) and (my_list[scan_pos] > key_value):
            my_list[scan_pos + 1] = my_list[scan_pos]
            scan_pos = scan_pos - 1

        # Everything's been moved out of the way, insert
        # the key into the correct location
        my_list[scan_pos + 1] = key_value

See an animation of the insertion sort at:

http://www.sorting-algorithms.com/insertion-sort

For another dance interpretation, search YouTube for “insertion sort dance” or use this link:

http://youtu.be/ROalU379l3U

You can trace through the code using Insertion Sort on Python Tutor.

21.4. Full Sorting Example

Full Sorting Example
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
import random


def selection_sort(my_list):
    """ Sort a list using the selection sort """

    # Loop through the entire array
    for cur_pos in range(len(my_list)):
        # Find the position that has the smallest number
        # Start with the current position
        min_pos = cur_pos

        # Scan left to right (end of the list)
        for scan_pos in range(cur_pos + 1, len(my_list)):

            # Is this position smallest?
            if my_list[scan_pos] < my_list[min_pos]:
                # It is, mark this position as the smallest
                min_pos = scan_pos

        # Swap the two values
        temp = my_list[min_pos]
        my_list[min_pos] = my_list[cur_pos]
        my_list[cur_pos] = temp


def insertion_sort(my_list):
    """ Sort a list using the insertion sort """

    # Start at the second element (pos 1).
    # Use this element to insert into the
    # list.
    for key_pos in range(1, len(my_list)):

        # Get the value of the element to insert
        key_value = my_list[key_pos]

        # Scan from right to the left (start of list)
        scan_pos = key_pos - 1

        # Loop each element, moving them up until
        # we reach the position the
        while (scan_pos >= 0) and (my_list[scan_pos] > key_value):
            my_list[scan_pos + 1] = my_list[scan_pos]
            scan_pos = scan_pos - 1

        # Everything's been moved out of the way, insert
        # the key into the correct location
        my_list[scan_pos + 1] = key_value


# This will point out a list
# For more information on the print formatting {:3}
# see the chapter on print formatting.
def print_list(my_list):
    for item in my_list:
        print(f"{item:3}", end="")
    print()


def main():
    # Create two lists of the same random numbers
    list_for_selection_sort = []
    list_for_insertion_sort = []
    list_size = 10
    for i in range(list_size):
        new_number = random.randrange(100)
        list_for_selection_sort.append(new_number)
        list_for_insertion_sort.append(new_number)

    # Print the original list
    print("Original List")
    print_list(list_for_selection_sort)

    # Use the selection sort and print the result
    print("Selection Sort")
    selection_sort(list_for_selection_sort)
    print_list(list_for_selection_sort)

    # Use the insertion sort and print the result
    print("Insertion Sort")
    insertion_sort(list_for_insertion_sort)
    print_list(list_for_insertion_sort)


main()