Decision Maths

1
Decision Maths

• Lesson 11 Sorting Algorithms

2
Sorting Algorithms

• Something as simple as sorting a list of numbers
  needs an algorithm on a computer.
• Look at the list of numbers below
• 8, 3, 5, 2, 7, 9, 1, 4, 6
• Clearly it is easy to see how the numbers would
  go from smallest to biggest.
• However a computer, does not have the ability to
  do this as simply as we can.
• It would use a sorting algorithm we are going to
  look at the following algorithms to sort these
  numbers
• Bubble Sort Algorithm
• Selection with interchange Sort Algorithm
• Quick Sort Algorithm
• Shuttle Sort Algorithm
• Insertion Sort Algorithm

3
Bubble Sort Algorithm

• We are going to sort out this list in to
  ascending order.
• The first thing to do is to compare the first two
  numbers to see if they are the correct way
  around.
• In this case they are not so we change them.
• Now repeat the process on the next pair of
  numbers, and so on.
• The process (or iteration) stops when all the
  pairs have been compared.
• The final line shows the State of the order after
  the first iteration.

• 1st Iteration

4
Bubble Sort Algorithm

• The process now begins again starting with the
  new state of the line.
• It is important to show all of your working and
  do not skip or assume any steps.
• Remember to show the state of the line at the end
  of each iteration.

• 2nd Iteration

5
Bubble Sort Algorithm

• The process now begins again starting with the
  new state of the line.
• It is important to show all of your working and
  do not skip or assume any steps.
• Remember to show the state of the line at the end
  of each iteration.

• 3rd Iteration

6
Bubble Sort Algorithm

• The process now begins again starting with the
  new state of the line.
• It is important to show all of your working and
  do not skip or assume any steps.
• Remember to show the state of the line at the end
  of each iteration.

• 4th Iteration

7
Bubble Sort Algorithm

• The process now begins again starting with the
  new state of the line.
• It is important to show all of your working and
  do not skip or assume any steps.
• Remember to show the state of the line at the end
  of each iteration.

• 5th Iteration

8
Bubble Sort Algorithm

• The process now begins again starting with the
  new state of the line.
• It is important to show all of your working and
  do not skip or assume any steps.
• Remember to show the state of the line at the end
  of each iteration.

• 6th Iteration

9
Bubble Sort Algorithm

• You will have seen from the last iteration that
  the final state of the line was in the correct
  order.
• You might think that this means you can stop the
  algorithm.
• Remember that an algorithm is a set or list of
  instructions that solve a particular problem.
• There is no intelligence within the algorithm to
  tell a computer that the correct answer has been
  reached.
• So you must run through the list one last time.
• The fact that no changes are made will tell the
  operator that the list must be in the correct
  order.

• 7th Iteration

10
Bubble Sort Algorithm

• Now see if you can complete the Algorithm with
  the questions below.
• Use the bubble sort algorithm to put the
  following numbers in ascending order.
• 8 6 9 2 5
• Use the bubble sort algorithm to put the
  following numbers in descending order.
• 8 6 9 2 5

11
Selection with InterchangeSort Algorithm

• In this Algorithm the smallest number is located
  and changed with the first.

12
Selection with InterchangeSort Algorithm

• The smallest number is now in the correct place,
  so you locate the second smallest and change it
  with the second number.

13
Selection with InterchangeSort Algorithm

• Then the third smallest gets swapped with the
  third number.
• The fourth smallest with the fourth number, and
  so on.

14
Selection with InterchangeSort Algorithm

• On the last pass you can see that the list has
  already been sorted.
• However, similar to previous algorithms all steps
  must be carried out to ensure the process works
  correctly.

15
Selection with InterchangeSort Algorithm

• Now see if you can complete the Algorithm with
  the questions below.
• Use the Selection with InterchangeSort Algorithm
  to put the following numbers in ascending order.
• 4 1 6 2 5 9
• Use the Selection with InterchangeSort Algorithm
  to put the following numbers in descending order.
• 4 1 6 2 5 9

16
Quick Sort Algorithm

• The quick sort Algorithm is based on a pivot
  number and sub-lists.
• The way of obtaining a pivot number can vary. In
  the Edexcel course they use a middle number (see
  very last slide).
• For this course we are going to use the first
  number in the list.
• Starting with your un-ordered list pick the pivot
  number.
• Here the pivot is the first number in the list,
  8.

17
Quick Sort Algorithm

• Now the pivot number is made permanent by placing
  everything bigger than it to the right and
  everything smaller to the left.
• This number is definitely in the correct place so
  it can be fixed.
• Notice how the order of the numbers has not
  changed.
• You now have two sub-lists either side of 8.
• Repeat the process of picking a pivot and sorting
  the list on each sub-list

8
9
8
18
Quick Sort Algorithm

• The pivot for the left list is 3.
• Fix the 3 as a permanent number and put
  everything bigger to the right and smaller to the
  left.
• Remembering not to change the order of the
  numbers.
• Once again we are certain that the 3 is in the
  correct place so we can fix its position.
• The pivot for the right list is the 9.
• As this is the only number in the sub-list then
  it to can be made fixed.

8
9
9
3
9
8
3
19
Quick Sort Algorithm

• We now have two sub-lists either side of the 3.
• For the left list the pivot is 2.
• Make it permanent.
• Arrange smaller numbers left and bigger numbers
  right.
• Fix 2 in final list.
• For the right list the pivot is 5.
• Make it permanent.
• Arrange smaller numbers left and bigger numbers
  right.
• Fix 5 in final list.

8
9
1
5
7
4
9
3
2
6
4
1
2
5
9
8
5
3
2
20
Quick Sort Algorithm

• There are now three sub-lists.
• Two of them are easily dealt with because both
  lists are single digits.
• So these can be fixed in the final list.
• The 7 is now the pivot in the last list.
• Make it permanent.
• The 6 moves across.
• We can fix the 7 in the final list.
• Now all thats left is the 6 so we can fix that
  too.

8
9
1
5
7
4
9
3
2
6
1
2
5
6
1
4
6
6
9
8
7
6
5
4
3
2
1
21
Quick Sort Algorithm

• Now see if you can complete the Algorithm with
  the questions below.
• Use the Selection with Quick Sort Algorithm to
  put the following numbers in ascending order.
• 4 1 6 2 5 9
• Use the Selection with Quick Sort Algorithm to
  put the following numbers in descending order.
• 4 1 6 2 5 9

22
The Shuttle Sort Algorithm

• Step 1 (first pass) compare the first two
  numbers,
• Swap if necessary to put numbers in ascending
  order.
• Step 2 (second pass) compare the second and
  third numbers
• Swap if necessary.
• Now compare the first and second again. Swapping
  if necessary.
• Step 3 (third pass) - compare the third and
  fourth numbers, then the second and third and
  then the first and second.

1st Pass
2nd Pass
3rd Pass
Original list
23
The Shuttle Sort Algorithm

• Step 4
• (4th pass)
• Compare the 4th and 3rd, 3rd and 2nd, 1st and
  2nd.
• Step 5
• (5th pass)
• Compare the 5th and 4th, 4th and 3rd, 3rd and
  2nd, 1st and 2nd.

4th Pass
5th Pass
24
The Shuttle Sort Algorithm

• Step 6
• (6th pass)
• Compare the 6th and 5th, 5th and 4th, 4th and
  3rd, 3rd and 2nd, 2nd and 1st.

6th Pass
25
The Shuttle Sort Algorithm

• Step 7
• (7th pass)
• Compare the 7th and 6th, 6th and 5th, 5th and
  4th, 4th and 3rd, 3rd and 2nd, 2nd and 1st.

7th Pass
26
The Shuttle Sort Algorithm

• Step 8
• (8th pass)
• Compare the 8th and 7th, 7th and 6th, 6th and
  5th, 5th and 4th, 4th and 3rd, 3rd and 2nd, 2nd
  and 1st.
• Hopefully you can see that this Algorithm will
  always have (n-1) iterations.

8th Pass
27
The Shuttle Sort Algorithm

• Now see if you can complete the Algorithm with
  the questions below.
• Use the Selection with The Shuttle Sort Algorithm
  to put the following numbers in ascending order.
• 3 8 6 8 1 9
• Use the Selection with The Shuttle Sort Algorithm
  to put the following numbers in descending order.
• 3 8 6 8 1 9

28
Insertion Sort Algorithm

• This is probably the most straight forward of the
  methods.
• Numbers should be taken one at a time from the
  original list, and inserted in their correct
  positions in the new list.

29
Insertion Sort Algorithm

• Again hopefully you can see that this will have n
  iterations.

30
Insertion Sort Algorithm

• Now see if you can complete the Algorithm with
  the questions below.
• Use the Selection with Insertion Sort Algorithm
  to put the following numbers in ascending order.
• 4 1 6 2 5 9
• Use the Selection with Insertion Sort Algorithm
  to put the following numbers in descending order.
• 4 1 6 2 5 9

31
Complexity

• Bubble Sort
• Selection with interchange
• Quick Sort
• Shuttle Sort
• Insertion Sort

32
Edexcel Quick Sort Algorithm
8
1
9
2
8
3
4
3
6
5
5
9
8
7
6
5
4
3
2
1