##### Laura Amatulli

(Rochester Hills - United States)Teaching 8th grade in a middle school in suburban Detroit I have strong interests in Earth Science and leadership. I have been a teacher consultant for our local National Writing Project site, Meadow Brook Writing Project, which keeps me active ...

## Algorithms - Lesson 5 - Different Types of Sorting

** Description:**This lesson explains the different types of sorting algorithms and how to differentiate them on the basis of time complexity, running time, space consumption, etc. Students can easily remember and memorize these things using the notes file attached inside this lesson.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

- Contributed By: NISHANT GUPTA

### Introduction to Sorting

** Description:**This lesson is a brief introduction about the sorting mechanism.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

**SORTING**

**What exactly does sorting mean?**

Sorting means to arrange the data in a certain order. There are many algorithms designed to serve this purpose. It is considered to be defacto problem for the study of algorithms. Let us see where exactly we can see the implementation of sorting.

Most of you have been googling for various stuff. Have you ever noticed the results that appear immediately in the drop down box of the google search toolbar when you hit any key on your keyboard. Yeah! That's sorting.. google sorts various results on its server for you. So basically what we have is

**INPUT****:** a sequence of n numbers (a1,a2,a3,......an)**OUTPUT:** A permutation of the numbers such that (a1 < a2 < a3 .... < an)

There are two types of sorting: internal and external sorting.

• **INTERNAL SORTING****:** If all the data to be stored can be accommodated at a time in memory then internal sorting methods are used. Various methods are discussed later in these notes.

• **External sorting****:** When the data to be stored is so large that some of the data has to be imported from auxiliary memory (hard disk,flash drive etc...), then external sorting methods are used. This is done by calling appropriate function at the time of sorting.

### Insertion Sort

** Description:**The insertion sort is a type of sorting method--one of the easiest and most widely used.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

**INSERTION SORT**

Different steps involved in the insertion sort are as follows:

**Step 1**: The very First element a[1] of any array is already sorted . So from the start itself we have two subsections of our array as shown below .

A: a[1] | a [2] | …….| a [k] | a[k+1] | ……. | a [n-1] | a [n]

LEFT .........................../...................................... RIGHT

The left portion of the array is sorted while RIGHT one (Elements a[2] .. a[n] ) is unsorted.

**Step 2**: Choose any element (usually the first one) from the RIGHT / Unsorted

Array portion then INSERT that element into the sorted LEFT portion in its proper position after shifting the elements of the LEFT portion (if needed). In this way the LEFT / SORTED portion increases by one

element while the RIGHT /UNSORTED portion decreases by one.

**Step 3**: Repeat STEP 2 until the LEFT portion covers the entire array.

Let's take an example to show the insertion sort. We need to sort (27,19,33,15,4)

Iteration | Num1 | Num2 | Num3 | Num4 | Num5 |
---|---|---|---|---|---|

Iteration1 | 27 | 19 | 33 | 15 | 4 |

Iteration2 | 19 | 27 | 33 | 15 | 4 |

Iteration2 | 19 | 27 | 33 | 15 | 4 |

Iteration3 | 19 | 27 | 33 | 15 | 4 |

Iteration4 | 15 | 19 | 27 | 33 | 4 |

Final Sorted | 4 | 15 | 19 | 27 | 33 |

*This is the iteration wise presentation for it.*

The colored numbers represent the numbers that are under consideration in the respective iteration.They represent the numbers that are compared.

**Pseudo code**

**INSERTION _SORT(A) Start1. For m= 2 to length [A] do2. lock = A[m]3. Arrange A[j] in sorted order4. n ← m -15. while n > 0 and A[n] > lock do6. A[n+1] = A[n]7. n = n-18. A[i+1] = key End **

**COMPLEXITY**

CASE | BEST CASE | AVERAGE CASE | WORST CASE |
---|---|---|---|

COMPLEXITY | n-1 | O(square of n) | O(square of n) |

**ADVANTAGES:**

1. Used most frequently by human beings in real life—ordering deck of cards

2. Efficient for small arrays.

3. Running time in best case is a function of n i.e linear as compared to SELECTION and BUBBLE sort which are quadratic in nature.

4. It uses no extra memory, it sorts in place.

### Selection Sort

** Description:**The selection sort is one type of sorting algorithm.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

**SELECTION SORT**

Let's take an example to sort a given list of numbers(27,19,33,15,4).

You must learn a few points before going through this iterative version of selection sort in an array. These are as follows:

The elements marked as yellow in any row are those that are compared at a particular time instant.The elements that are marked with green are those elements that had already been sorted until the starting of the iteration or left out as sorted until that particular iteration ends.

**First Iteration**

Num1 | Num2 | Num3 | Num4 | Num5 | ||||
---|---|---|---|---|---|---|---|---|

27 | 19 | 33 | 15 | 4 | ||||

19 | 27 | 33 | 15 | 4 | ||||

19 | 27 | 33 | 15 | 4 | ||||

15 | 27 | 33 | 19 | 4 4 | 27 | 33 | 19 | 15 |

**Second Iteration**

Num1 | Num2 | Num3 | Num4 | Num5 |
---|---|---|---|---|

4 | 27 | 33 | 19 | 15 |

4 | 27 | 33 | 19 | 15 |

4 | 19 | 33 | 27 | 15 |

4 | 15 | 33 | 27 | 19 |

**Third Iteration**

Num1 | Num2 | Num3 | Num4 | Num5 |
---|---|---|---|---|

4 | 15 | 33 | 27 | 19 |

4 | 15 | 27 | 33 | 19 |

4 | 15 | 19 | 33 | 27 |

**Fourth Iteration**

Num1 | Num2 | Num3 | Num4 | Num5 |
---|---|---|---|---|

4 | 15 | 19 | 33 | 27 |

4 | 15 | 19 | 27 | 33 |

*this is the iteration wise presentation of sorting step by step*

This is the best way to understand this topic.

**Pseudo Code**

**STARTSET l=size of array For(i=0;i FOR(j=i;j If (A[i]>A[j]) Swap(A[i],A[j]) End if statement End for loop End for loopEND**

**COMPLEXITY**

CASE | BEST CASE | AVERAGE CASE | WORST CASE |
---|---|---|---|

COMPLEXITY | O(square of n) | O(square of n) | O(square of n) |

These are very general pseudo codes which helps a lot to develop the actual working C/c++ codes . For exact code you can refer to this link - click here

### Quick Sort

** Description:**The quick sort is a sorting mechanism based on the divide and conquer method.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

**Quick sort**

Quick sort was discovered by Tony
Hoare in 1962. In this algorithm, the hard work is splitting the array
into subsets so that merging the final result is trivial.

**1. Choose a pivot element (it can be any element but preferably) from the array.2.**

**Split the array into three sub-arrays containing the items less than**

**the pivot, the pivot itself, and the items bigger than the pivot.**

3. Recursively calling (quick sort) function on the first and last sub-array.

3. Recursively calling (quick sort) function on the first and last sub-array.

This is exactly what we do:

We put the pivot element on its right
position and then again divide the array into two sub-arrays with
reference to that pivot element. The newly generated sub-arrays will
have their own new pivot elements. The choice of the pivot element is
completely user oriented. And then again through the recursive
mechanism these sub-arrays are called in which again their pivot
elements are put on their respective position and that array is
further divided again. This process continues until we have a
single element left in the sub-arrays, which is in itself in its right
position.

Let me present a pictorial example to explain quick sort.

We have to sort a given list **(11,10,8,12,4,3,9)**.

So the different steps that come while under the quick sort Algorithm that use the divide and conquer technique is as follows:

Here's a more formal specification of the quick sort algorithm. The separate partition subroutine takes the original position of the pivot element as input and returns the post-partition pivot position as output.

**Quicksort(A, p, r):{ if p < r: q = Partition(A, p, r) // calling partion function which returns the bifurcation point Quicksort(A, p, q) // calling quicksort for the first array Quicksort(A, q+1, r) // calling quicksort for the second array}Partition(A, p, r): { x = A[p] // considering the first element to ba as the pivot element i = p - 1 j = r + 1 while 1: j = j - 1 while A[j] <= x: j = j - 1 i = i + 1 while A[i] >= x: j = j - 1 if i < j: swap(A[i], A[j]) else: return j}**

In the above code (

**Quick sort**) we are inputting an array containing the numbers which are to be sorted with 'n' as the total number of the numbers present in it. The function (

**Partition**) returns a position about which that array/sub-array is to be partitioned again. This returned value is again used in the recursive calls made by the function (

**Quick sort**).

Lets see the complexity expression for quick sort

CASE | BEST CASE | AVERAGE CASE | WORST CASE |
---|---|---|---|

COMPLEXITY | O(n logn) | O(n logn) | O(square of n) |

And in order to remove this problem with quick sort, a new sorting called Randomized Quick sortis made. The only difference is that the pivot element is chosen randomly rather than selecting the same one position of the array as the pivot element.

Lets see the complexity expression for Randomized Quick Sort

CASE | BEST CASE | AVERAGE CASE | WORST CASE |
---|---|---|---|

COMPLEXITY | O(n logn) | O(n logn) | O(n logn) |

### Heap Sort

** Description:**The heap sort involves the TREE (data structure) as the basic element to store a list.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Lesson Plan

**HEAP SORT**

Heap Sort involves different functions. They are provided here in detail. I hope you will enjoy it.

**MAX-HEAP(A,i)**

1. l <- Left(i)

2. r<- Right(i)

3. if l≤ heapsize[A] and A[l]>A[i]

4. then largest<-l

5. else largest<-i

6. if r≤heap-size [A] and A[r]>A[largest]

7. then largest<-r

8. if largest not equal to i

9. then exchange A[i]↔A[largest]

10. MAX-HEAP(A,largest)

At each step, the largest of the elements A[i], A[LEFT], A[RIGHT] are determined and its index is stored in largest. Then the sub-tree rooted at node i is a max-heap and the procedure terminates. Otherwise, one of the two children has the largest element, and A[i] is swapped with A[largest], which causes node i and its children to satisfy the max-heap property. The node indexed by largest, however, now has the original value A[i],and thus the sub-tree rooted at largest may violate the max-heap property. So, max-heap must be called recursively on that sub-tree.

**BUILD-MAX HEAP(A)**

1. Heapsize[A] <-length[A]

2. For i<-[length[A]/2 downto 1

3. Do MAXHEAP(A,i)

**HEAPSORT(A)**

1. BUILD-MAX-HEAP(A)

2. For i<-length[A] downto 2

3. Do exchange A[1]↔A[i]

4. Heapsize[A] <- heapsize[A]-1

5. MAXHEAP(A,1)

FUNCTION | MAX-HEAP | BUILD-MAX HEAP | HEAP SORT |
---|---|---|---|

COMPLEXITY | O(n) | O(log n) | O(n log n) |

So Here we complete the Heap-sort Algorithm.

For the complete C code You can refer this link which is uploaded by some other member.

### Sorting Presentation

** Description:**This presentation includes a brief summary about the sorts discussed in this lesson.

**Last Updated:**

**Subject(s):**- Career & Technical Education
- Mathematics
- ...

**Educational Level(s):**- Grades 11-12 / Ages 16-18
- College & Beyond

- high
- 11th
- 12th
- secondary
- senior
- teen

**Instructional Component Type(s):**- Curriculum: Study Guide/Notes

This presentation includes a brief summary about the sorts discussed in this lesson.