sorting algorithms v01

Sorting Algorithms

2

Instructions

• For the embedded lessons look for the Icon

Click on it to present embedded lessons

Lecture Objectives

• Become aware of what is sorting. • Become aware of sorting algorithms.• Explore the types of sorting algorithms. • Learn about Bubble Sort.• Learn about Insertion Sort.• Learn about Selection Sort.• Learn about Merge Sort.

What is sorting?

An operation that puts (organizes) elements of a list (a collection of data) into certain order (ascending or descending order).

Sorting = ordering.Sorted = ordered based on a particular

way. 3 1 6 2 1 3 4 5 9 0

0 1 1 2 3 3 4 5 6 9

9 6 5 4 3 3 2 1 1 0

Ascending or descending order

Sorting – formal definition

Input: A sequence of n numbers: a1, a2, ..., an

Output: A permutation (reordering):

ak1, ak2, ..., akn

of the input sequence such that

f(ak1) f(ak2) ... f(akn)

where f is an ordering function

Permutation

Examples of Sorting

– Words in a dictionary are sorted. – Files in a directory are often listed in sorted order.– In a newspaper, the calendar of events in a schedule is generally

sorted by date.– In a record store musical compact disks are generally sorted by

recording artist.

What is a Sorting Algorithm?An algorithm for sorting a set of elements.

3 1 6 2 1 3 4 5 9 0

0 1 1 2 3 3 4 5 6 9

9 6 5 4 3 3 2 1 1 0

Types of Sorting Algorithms

There are many, many different types of sorting algorithms, but the primary ones are:

• Bubble Sort• Selection Sort• Insertion Sort• Merge Sort• Quick Sort• Shell Sort

Selection Sort

One of the simplest sorting algorithms works as follows:

– Find the smallest element and exchange it with the element in the first position

– Find the second smallest element and exchange it with the element in the second position

– Continuing in this way until the entire array is sorted.

Name “selection sort” because it works by repeatedly “selecting“ the smallest remaining

element and exchanging it with the i-th element in i-th iteration.

Meaning of words

5 1 3 4 6 2

Comparison

Data Movement

Sorted

Selection Sort

5 1 3 4 6 2

Comparison

Data Movement

Sorted

Largest

Selection Sort

5 1 3 4 2 6

Comparison

Data Movement

Sorted

Selection Sort

5 1 3 4 2 6

Comparison

Data Movement

Sorted

Largest

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Largest

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Largest

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Selection Sort

2 1 3 4 5 6

Comparison

Data Movement

Sorted

Largest

Selection Sort

1 2 3 4 5 6

Comparison

Data Movement

Sorted

Selection Sort

1 2 3 4 5 6

Comparison

Data Movement

Sorted

DONE!

Selection Sort

46

start

i:=1,n-1

min:=a(i)ind:=i

for j:=i+1,n

a(j) < min

min:=a(j)ind:=j

a(ind):=a(i)a(i):=min

end

false

Selection sort

Code of selection sortpublic static void selectionSort(int[] a) {

int outer, inner, min; for (outer = 0; outer < a.length - 1; outer++)

{ min = outer;

for (inner = outer + 1; inner < a.length; inner++) {

if (a[inner] < a[min]) min = inner;} // Invariant: for all i, if outer <= i <= inner, then a[min] <= a[i]

// a[min] is least among a[outer]..a[a.length - 1] int temp = a[outer]; a[outer] = a[min]; a[min] = temp;// Invariant: for all i <= outer, if i < j then a[i] <= a[j] }

}

1. We have two group of items:–sorted group, and–unsorted group

2. Initially, all items in the unsorted group and the sorted group is empty. –We assume that items in the unsorted group unsorted. –We have to keep items in the sorted group sorted.

3. Pick any item from, then insert the item at the right position in the sorted group to maintain sorted property.4. Repeat the process until the unsorted group becomes empty.

Insertion sort

This method is often used for playing cards (used by people to sort bridge hands).

Like sorting a hand of playing cards:–Start with an empty left hand and the cards facing down on the table.–Remove one card at a time from the table, and insert it into the correct position in the left hand

Compare it with each of the cards already in the hand, from right to left.–The cards held in the left hand are sorted

These cards were originally the top cards of the pile on the table.

Insertion sort

50

6 1024

12

36

Insertion sort

1. To insert 12, we need to make room for it.

51

6 1024 36

12

Insertion sort

2. Moving 36.

52

6 10 24 36

12

Insertion sort

3. Moving 24.

53

6 10 243612

Insertion sort

4. Place 12 into right position.

for(i=1; i<n; i++)for(j=i; (j>0) && (a[j]>a[j-1]); j--){

pom=a[j];a[j]=a[j-1];a[j-1]=pom;

}

Code for insertion sort

Insertion sortwith Romanian folk dance

• Compare each element (execept the last one) with its neighbor to the right.– If they are out of order, swap them.– This puts the largest element at the very end.– The last element is now in the correct and final place.

• Compare each element (execept the last two) with its neighbor to the right.– If they are out of order, swap them.– This puts the largest element at the very end.– The last two elements is now in the correct and final place.

• Compare each element (execept the last three) with its neighbor to the right.– Continue as above until you have no unsorted elements on the left.

Bubble sort

9, 6, 2, 12, 11, 9, 3, 7

6, 9, 2, 12, 11, 9, 3, 7

6, 2, 9, 12, 11, 9, 3, 7In the third comparison, the 9 is not larger than the 12 so no exchange is made. We move on to compare the next pair without any change to the list.

In the third comparison, the 9 is not larger than the 12 so no exchange is made. We move on to compare the next pair without any change to the list.

The next pair of numbers are compared. 9 is the larger and this pair is also exchanged.The next pair of numbers are compared. 9 is the larger and this pair is also exchanged.

Compares the numbers in pairs from left to right exchanging when necessary. The first number is compared to the second and as it is larger they are exchanged.

Compares the numbers in pairs from left to right exchanging when necessary. The first number is compared to the second and as it is larger they are exchanged.

Bubble sort

6, 2, 9, 12, 11, 9, 3, 7

6, 2, 9, 11, 12, 9, 3, 7

6, 2, 9, 11, 9, 12, 3, 7The 12 is greater than the 3 so they are exchanged.The 12 is greater than the 3 so they are exchanged.

The 12 is greater than the 9 so they are exchangedThe 12 is greater than the 9 so they are exchanged

The 12 is larger than the 11 so they are exchanged.The 12 is larger than the 11 so they are exchanged.

Bubble sort

6, 2, 9, 11, 9, 3, 12, 7

6, 2, 9, 11, 9, 3, 7, 12

The 12 is greater than the 7 so they are exchanged.The 12 is greater than the 7 so they are exchanged.

The end of the list has been reached so this is the end of the first pass. The twelve at the end of the list must be largest number in the list and so is now in the correct position. We now start a new pass from left to right.

The end of the list has been reached so this is the end of the first pass. The twelve at the end of the list must be largest number in the list and so is now in the correct position. We now start a new pass from left to right.

Bubble sort

6, 2, 9, 11, 9, 3, 7, 122, 6, 9, 11, 9, 3, 7, 122, 6, 9, 9, 11, 3, 7, 122, 6, 9, 9, 3, 11, 7, 122, 6, 9, 9, 3, 7, 11, 12

6, 2, 9, 11, 9, 3, 7, 12

This time we do not have to compare the last two numbers as we know the 12 is in position. This pass therefore only requires 6 comparisons.This time we do not have to compare the last two numbers as we know the 12 is in position. This pass therefore only requires 6 comparisons.

First Pass

Second Pass

Bubble sort

2, 6, 9, 9, 3, 7, 11, 122, 6, 9, 3, 9, 7, 11, 122, 6, 9, 3, 7, 9, 11, 12

6, 2, 9, 11, 9, 3, 7, 12

2, 6, 9, 9, 3, 7, 11, 12Second Pass

First Pass

Third Pass

This time the 11 and 12 are in position. This pass therefore only requires 5 comparisons.This time the 11 and 12 are in position. This pass therefore only requires 5 comparisons.

Bubble sort

2, 6, 9, 3, 7, 9, 11, 122, 6, 3, 9, 7, 9, 11, 122, 6, 3, 7, 9, 9, 11, 12

6, 2, 9, 11, 9, 3, 7, 12

2, 6, 9, 9, 3, 7, 11, 12Second Pass

First Pass

Third Pass

Each pass requires fewer comparisons. This time only 4 are needed.Each pass requires fewer comparisons. This time only 4 are needed.

2, 6, 9, 3, 7, 9, 11, 12Fourth Pass

Bubble sort

2, 6, 3, 7, 9, 9, 11, 122, 3, 6, 7, 9, 9, 11, 12

6, 2, 9, 11, 9, 3, 7, 12

2, 6, 9, 9, 3, 7, 11, 12Second Pass

First Pass

Third Pass

The list is now sorted but the algorithm does not know this until it completes a pass with no exchanges.The list is now sorted but the algorithm does not know this until it completes a pass with no exchanges.

2, 6, 9, 3, 7, 9, 11, 12Fourth Pass2, 6, 3, 7, 9, 9, 11, 12

Fifth Pass

Bubble sort

2, 3, 6, 7, 9, 9, 11, 12

6, 2, 9, 11, 9, 3, 7, 12

2, 6, 9, 9, 3, 7, 11, 12Second Pass

First Pass

Third Pass2, 6, 9, 3, 7, 9, 11, 12Fourth Pass2, 6, 3, 7, 9, 9, 11, 12

Fifth Pass

Sixth Pass

2, 3, 6, 7, 9, 9, 11, 12This pass no exchanges are made so the algorithm knows the list is sorted. It can therefore save time by not doing the final pass. With other lists this check could save much more work.

This pass no exchanges are made so the algorithm knows the list is sorted. It can therefore save time by not doing the final pass. With other lists this check could save much more work.

Bubble sort

65

start

i=2,n

j=n,i,-1

a(j-1) > a(j)

temp=a(j-1)a(j-1)=a(j)a(j)=temp

end

false

Bubble sort

Code for bubble sort

public static void bubbleSort(int[] a) { int outer, inner; for (outer = a.length - 1; outer > 0; outer--)

{ // counting down for (inner = 0; inner < outer; inner++)

{ // bubbling up if (a[inner] > a[inner + 1])

{ // if out of order... int temp = a[inner];

// ...then swap a[inner] = a[inner + 1]; a[inner + 1] = temp; } } }}

67

Bubble sort with Hungarian ("Csángó") folk dance

• Divide array into two halves.• Recursively sort each half.• Merge two halves to make sorted whole.

merge

sort

A L G O R I T H M S

divideA L G O R I T H M S

A G L O R H I M S T

A G H I L M O R S T

Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

• Keep track of smallest element in each sorted half.• Insert smallest of two elements into auxiliary array.• Repeat until done.

A

Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G

Merge sort


auxiliary array

smallest smallest

A G L O R H I M S T

A G H


Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I


Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L


Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L M


Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L M O


Merge sort

auxiliary array

smallest smallest

A G L O R H I M S T

A G H I L M O R


Merge sort

auxiliary array

first halfexhausted smallest

A G L O R H I M S T

A G H I L M O R S


Merge sort

auxiliary array

first halfexhausted smallest

A G L O R H I M S T

A G H I L M O R S T


Merge sort

Merge sort

We learnt!

• What is sorting. • Sorting algorithms.• Types of sorting algorithms. • Bubble Sort.• Insertion Sort.• Selection Sort.• Merge Sort.

Quiz

A

B

C

D

1. One of the simplest sorting algorithms is called bubble sort. Do you know why?

It encases each element in a 'bubble' before sorting them.

It's a mystery. Why is anything called what it is, really?The designer hoped more people would use his sort if it had a cute nameSmaller elements 'bubble' to the top

Teacher Guide - Quiz Question

Teacher Guide - Quiz Solution

Quiz

A

B

C

D

2.In which case would an insertion sort perform best, assuming it was reading the array to be sorted from beginning to end (as opposed to randomly)?

If the array was sorted in reverse order.

If the array was already sorted.

If the array was in a random order.

These will all perform equally well.

Sort the following array using each of the four sorting algorithms.

26 48 12 92 28 6 33

Homework

a) Bubble Sortb) Selection Sortc) Insertion Sortd) Merge Sort

sorting algorithms v01

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