Bubble sort is a simple sorting algorithm. This sorting algorithm is comparison-based algorithm in which each pair of adjacent elements is compared and the elements are swapped if they are not in order. This algorithm is not suitable for large data sets as its average and worst case complexity are of Ο(n2) where n is the number of items.
How Bubble Sort Works?
We take an unsorted array for our example. Bubble sort takes Ο(n2) time so we're keeping it short and precise.
Bubble sort starts with very first two elements, comparing them to check which one is greater.
In this case, value 33 is greater than 14, so it is already in sorted locations. Next, we compare 33 with 27.
We find that 27 is smaller than 33 and these two values must be swapped.
The new array should look like this −
Next we compare 33 and 35. We find that both are in already sorted positions.
Then we move to the next two values, 35 and 10.
We know then that 10 is smaller 35. Hence they are not sorted.
We swap these values. We find that we have reached the end of the array. After one iteration, the array should look like this −
To be precise, we are now showing how an array should look like after each iteration. After the second iteration, it should look like this −
Notice that after each iteration, at least one value moves at the end.
And when there's no swap required, bubble sorts learns that an array is completely sorted.
Now we should look into some practical aspects of bubble sort.
Algorithm
We assume list is an array of n elements. We further assume that swap function swaps the values of the given array elements.
begin BubbleSort(list) for all elements of list if list[i] > list[i+1] swap(list[i], list[i+1]) end if end for return list end BubbleSort
Pseudocode
We observe in algorithm that Bubble Sort compares each pair of array element unless the whole array is completely sorted in an ascending order. This may cause a few complexity issues like what if the array needs no more swapping as all the elements are already ascending.
To ease-out the issue, we use one flag variable swapped which will help us see if any swap has happened or not. If no swap has occurred, i.e. the array requires no more processing to be sorted, it will come out of the loop.
Pseudocode of BubbleSort algorithm can be written as follows −
procedure bubbleSort( list : array of items ) loop = list.count; for i = 0 to loop-1 do: swapped = false for j = 0 to loop-1 do: /* compare the adjacent elements */ if list[j] > list[j+1] then /* swap them */ swap( list[j], list[j+1] ) swapped = true end if end for /*if no number was swapped that means array is sorted now, break the loop.*/ if(not swapped) then break end if end for end procedure return list
Implementation
One more issue we did not address in our original algorithm and its improvised pseudocode, is that, after every iteration the highest values settles down at the end of the array. Hence, the next iteration need not include already sorted elements. For this purpose, in our implementation, we restrict the inner loop to avoid already sorted values.
This is an in-place comparison-based sorting algorithm. Here, a sub-list is maintained which is always sorted. For example, the lower part of an array is maintained to be sorted. An element which is to be 'insert'ed in this sorted sub-list, has to find its appropriate place and then it has to be inserted there. Hence the name, insertion sort.
The array is searched sequentially and unsorted items are moved and inserted into the sorted sub-list (in the same array). This algorithm is not suitable for large data sets as its average and worst case complexity are of Ο(n2), where n is the number of items.
How Insertion Sort Works?
We take an unsorted array for our example.
Insertion sort compares the first two elements.
It finds that both 14 and 33 are already in ascending order. For now, 14 is in sorted sub-list.
Insertion sort moves ahead and compares 33 with 27.
And finds that 33 is not in the correct position.
It swaps 33 with 27. It also checks with all the elements of sorted sub-list. Here we see that the sorted sub-list has only one element 14, and 27 is greater than 14. Hence, the sorted sub-list remains sorted after swapping.
By now we have 14 and 27 in the sorted sub-list. Next, it compares 33 with 10.
These values are not in a sorted order.
So we swap them.
However, swapping makes 27 and 10 unsorted.
Hence, we swap them too.
Again we find 14 and 10 in an unsorted order.
We swap them again. By the end of third iteration, we have a sorted sub-list of 4 items.
This process goes on until all the unsorted values are covered in a sorted sub-list. Now we shall see some programming aspects of insertion sort.
Algorithm
Now we have a bigger picture of how this sorting technique works, so we can derive simple steps by which we can achieve insertion sort.
Step 1 − If it is the first element, it is already sorted. return 1; Step 2 − Pick next element Step 3 − Compare with all elements in the sorted sub-list Step 4 − Shift all the elements in the sorted sub-list that is greater than the value to be sorted Step 5 − Insert the value Step 6 − Repeat until list is sorted
Pseudocode
procedure insertionSort( A : array of items ) int holePosition int valueToInsert for i = 1 to length(A) inclusive do: /* select value to be inserted */ valueToInsert = A[i] holePosition = i /*locate hole position for the element to be inserted */ while holePosition > 0 and A[holePosition-1] > valueToInsert do: A[holePosition] = A[holePosition-1] holePosition = holePosition -1 end while /* insert the number at hole position */ A[holePosition] = valueToInsert end for end procedure
Selection sort is a simple sorting algorithm. This sorting algorithm is an in-place comparison-based algorithm in which the list is divided into two parts, the sorted part at the left end and the unsorted part at the right end. Initially, the sorted part is empty and the unsorted part is the entire list.
The smallest element is selected from the unsorted array and swapped with the leftmost element, and that element becomes a part of the sorted array. This process continues moving unsorted array boundary by one element to the right.
This algorithm is not suitable for large data sets as its average and worst case complexities are of Ο(n2), where n is the number of items.
How Selection Sort Works?
Consider the following depicted array as an example.
For the first position in the sorted list, the whole list is scanned sequentially. The first position where 14 is stored presently, we search the whole list and find that 10 is the lowest value.
So we replace 14 with 10. After one iteration 10, which happens to be the minimum value in the list, appears in the first position of the sorted list.
For the second position, where 33 is residing, we start scanning the rest of the list in a linear manner.
We find that 14 is the second lowest value in the list and it should appear at the second place. We swap these values.
After two iterations, two least values are positioned at the beginning in a sorted manner.
The same process is applied to the rest of the items in the array.
Following is a pictorial depiction of the entire sorting process −
Now, let us learn some programming aspects of selection sort.
Algorithm
Step 1 − Set MIN to location 0 Step 2 − Search the minimum element in the list Step 3 − Swap with value at location MIN Step 4 − Increment MIN to point to next element Step 5 − Repeat until list is sorted
Pseudocode
procedure selection sort list : array of items n : size of list for i = 1 to n - 1 /* set current element as minimum*/ min = i /* check the element to be minimum */ for j = i+1 to n if list[j] < list[min] then min = j; end if end for /* swap the minimum element with the current element*/ if indexMin != i then swap list[min] and list[i] end if end for end procedure
Merge sort is a sorting technique based on divide and conquer technique. With worst-case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted manner.
How Merge Sort Works?
To understand merge sort, we take an unsorted array as the following −
We know that merge sort first divides the whole array iteratively into equal halves unless the atomic values are achieved. We see here that an array of 8 items is divided into two arrays of size 4.
This does not change the sequence of appearance of items in the original. Now we divide these two arrays into halves.
We further divide these arrays and we achieve atomic value which can no more be divided.
Now, we combine them in exactly the same manner as they were broken down. Please note the color codes given to these lists.
We first compare the element for each list and then combine them into another list in a sorted manner. We see that 14 and 33 are in sorted positions. We compare 27 and 10 and in the target list of 2 values we put 10 first, followed by 27. We change the order of 19 and 35 whereas 42 and 44 are placed sequentially.
In the next iteration of the combining phase, we compare lists of two data values, and merge them into a list of found data values placing all in a sorted order.
After the final merging, the list should look like this −
Now we should learn some programming aspects of merge sorting.
Algorithm
Merge sort keeps on dividing the list into equal halves until it can no more be divided. By definition, if it is only one element in the list, it is sorted. Then, merge sort combines the smaller sorted lists keeping the new list sorted too.
Step 1 − if it is only one element in the list it is already sorted, return.
Step 2 − divide the list recursively into two halves until it can no more be divided.
Step 3 − merge the smaller lists into new list in sorted order.
Pseudocode
We shall now see the pseudocodes for merge sort functions. As our algorithms point out two main functions − divide & merge.
Merge sort works with recursion and we shall see our implementation in the same way.
procedure mergesort( var a as array )
if ( n == 1 ) return a
var l1 as array = a[0] ... a[n/2]
var l2 as array = a[n/2+1] ... a[n]
l1 = mergesort( l1 )
l2 = mergesort( l2 )
return merge( l1, l2 )
end procedure
procedure merge( var a as array, var b as array )
var c as array
while ( a and b have elements )
if ( a[0] > b[0] )
add b[0] to the end of c
remove b[0] from b
else
add a[0] to the end of c
remove a[0] from a
end if
end while
while ( a has elements )
add a[0] to the end of c
remove a[0] from a
end while
while ( b has elements )
add b[0] to the end of c
remove b[0] from b
end while
return c
end procedure
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