Radix sort

Radix sort

            Let a be an array of n numbers. Our aim is to sort this array in ascending order using radix sort. The steps to be followed are

i)                    Initialize ten queues namely q[0], q[1], ………, q[9]. That is set rear [i] = front [i] = NULL for all I = 0, 1, 2, ……,9.

ii)                  Scan the array from left to right and find the least significant digit for all array elements.

iii)                If it is 0, push the number in q[0]; if it is 1, push the number in q[1],….., and if it is 9, push the number in q[9].

iv)                After pushing all the elements in the respective queues, pop up the data from 0^th queue to 9^th queue and restore in the array a.

v)                  Again scan the array from left to right and find the second least significant digit for all array elements.

vi)                Repeats steps (iii) and (iv)

The repetition of the above process depends upon the width of the maximum element in the array. That is if the maximum width of the array element is 3, then scan the array three times and find the 1^st least significant digit and 3^rd least significant digit.

Example

Merge sort

Merge sort

            Let a be an array of n numbers. Our aim is to sort this array in ascending order using merge sort. The steps to be followed are

i)                    Divide the array into n sub arrays with size 1.

ii)                  Merge adjacent pair of sub arrays [a[1]] and [a[2]], [a[3]] and [a[4]], …….,

[a[n-1]] and [a[n]]. Now we will get a n/2 sorted sub arrays of size 2 or less.

iii)                Merge adjacent pairs of sub arrays [a[1]] and [a[2]], [a[3]] and [a[4]], ……., [a[n-3]], [a[n-2]] and [a[n-1

iv)                Repeat this process until ther]], [a[n]]. Now we will get a n/4 sorted sub arrays of size 4 or less.e is only one sub array of size n.

Now this array will be in sorted order

Example

Merge

Merge

            Merging is a process of combining two or more sorted arrays into a single sorted array.

Explanation

            Suppose A[m] and B[n] are two sorted arrays. Merging is a process of combining these two arrays into a single sorted array C[m+n]. Steps to be followed are

i)                    Scan both arrays left to right

ii)                  Compare A[1] with B[1]. If A[1] is less than B[1] put A[1] as C[1] and compare A[2] with B[1] and so on, else put B[1] as C[1] and compare A[1] with B[2] and so on.

iii)                Repeat step ii until any one array becomes empty.

iv)                If A array becomes empty, store all the remaining B array elements in the same order in the C array. Besides store all the remaining A array elements in the same order in the B array.

Procedure

Selection sort

Selection sort

            Let A be an array of n elements A[1], A[2], …., A[n]. Our aim is to sort the numbers in the array in ascending order. The steps are as follows

1)      Find the smallest element position in the list A[1], A[2], …., A[n] say k₁ and interchange the values A[1] and A[k₁]

2)      Find the smallest element position in the list A[2], A[3], ….., A[n] say k₂ and interchange the values A[2] and A[k₂].

3)      Find the smallest element position in the list A[3], A[4], ….., A[n] say k₃ and then interchange the values A[3] and A[k₃]

4)      Find the smallest element position in the list A[n-1] and A[n] say k_n-1 and then interchange the values A[n] and A[k_n-1].

Example -        Sort the array A using selection sort

                        A = (43, 72, 10, 23, 80, 1, 75)

           

Insertion sort

Insertion sort

            Let a be an array of numbers a[1], a[2] ….. a[n]. Our aim is to sort them in ascending order using insertion sort.

Principle

            Scan the array a from a[1] to a[n] and find the smallest elements a[r], r = 1, 2 ……, r and insert into its proper position in the previously sorted sub array a[1], a[2] ….  a[r]. This can be explained by the following steps.

i)                    For r = 1 the previously sorted such array is empty therefore a[1] by itself is sorted.

ii)                  For r =2 , a[2] is inserted into the proper position in the previously sorted sub array a[1] i.e.; a[2] is inserted after or before a[1].

iii)                For r = 3, a[3] is inserted into the proper position in the previously sorted sub array a[1], a[2]. i.e.; a[3] is inserted either before a[1] or after a[2] or in between a[1] and a[2].

The above steps are repeated until the last element a[n] is inserted into its proper position in the sorted sub array. Now the given array a is in sorted order.

Example

Bubble sort

Bubble sort

            Let a be an array of n numbers. Our aim is to sort the numbers in the array in ascending order. The principle is to scan the array sequentially n-1 times and in each scan do the following steps.

Scan – 1

i)                    Take the first element and compare it with the second element. If it is small don’t interchange; or else interchange the two elements.

ii)                  Then compare the second element with the third element. If it is small don’t interchange; or else interchange the two elements.

iii)                This process is continued till (n-1)^th element is compared with the n^th element.

iv)                At the end of the first scan, the highest element comes to the n^th position.

Scan – 2

SORTING

1 Sorting

General Concepts

            A file is an ordered collection of items.Each item in the file is called a record. The records are named as r[1], r[2],…..r[n], where n is the size of the file.

            A key is associated with each record r[i] and through this key only we can identify the records in the file.