Shell Sort

Shell Sort: Shell sort is a sorting algorithm. It is an extended version of the insertion sort.

• In this sorting, we compare the elements that are distant apart rather than the adjacent.
• We start by comparing elements that are at a certain distance apart. So, if there are N elements then we start with a value gap < N.
• In each iteration we keep decreasing the value of the gap until the difference becomes 1.
• In the last iteration, the shell sort acts as an insertion-type.

Complexity table of shell sort

Complexity	Best case	Average case	Worst case
Time	O(n)	O((nlog(n))2)	O((nlog(n))2)
Space			O(1)

Algorithm of shell sort

Shell_Sort (array, size)
for interval i ? size /2 down to 1.     
for each interval "i" in the array.        
sort all the elements at interval "i".  
end Shell_Sort.  

Shell sort example

Suppose you have the following array, which you have to sort.

14

18

19

37

23

40

29

30

11

The size of array is 9, so n = 9.

    Now, find the value of the Gap.                             

Gap = floor(n / 2)                                     

= floor (9 / 2)                                     

= floor(4.5)                                     

= 4                

Pass = 1 Gap = 4  

Gap 4 means, if we consider the first element at index 0 then next element will be at index 0+4 = 4., and third element will be at index 4 + 4 = 8.

 Similarly, if we consider the element at index 1 then next element will be at index 1 + 4 = 5, and third element will be at index 5 + 4 = 9. But there is no 9^th index in the above array so we will just have index 1 and 5.

Similarly, if we consider the element at index 2, 3, and 4 respectively then next element will be at index 2 + 4 = 6, 3 + 4 = 7, and 4 + 4 = 8 respectively.

  Compare the elements.  

A[0] > A[4] = 14 > 23 // false  

A[1] > A[5] = 18 > 40 // false  

A[2] > A[6] = 19 > 29 // false  

A[3] > A[7] = 37 > 30 // true i.e., swap the both element positions.

A[4] > A[8] = 23 > 11 // true i.e., swap the both element positions.  

A[1] > A[4] = 14 > 11 // true i.e., swap the both element positions.

Pass 1 is complete.  

Pass 2: Gap = floor(Gap / 2)            

Gap = floor(4 / 2)  // in pass 1,

Gap was 4.            

Gap = 2  

Gap 2 means, if we consider the first element at index 0 then next element will be at index 0+2 = 2, and similarly this process applied till the end of index.

Compare the elements.

A[0] > A[2] = 11 > 19 // false

A[]1 > A[3] = 18 > 30 // false

A[2] > A[4] = 19 > 14 // true, i.e. swap the both element positions.

A[1] > A[2] = 11 > 14 // false

A[3] > A[5] = 30 > 40 //false

A[4] > A[6] = 19 > 29 // false

A[5] > A[7] = 40 > 37 // true, i.e. swap the both element positions.

A[3] > A[5] = 30 > 37 // false

A[6] > A[8] = 29 > 23 // true, i.e. swap the both element positions.

A[4] > A[6] = 19 > 23 // false

Pass 2 is complete.  

Pass 3: Gap = floor(Gap/2)             

Gap = floor(2 / 2)  // in pass 2, Gap was 2.             

Gap = 1                 // i.e. last pass when gap is 1, shell sort is like the insertion sort. Compare the elements.

A[0] > A[1] = 11 > 18 // false

A[1] > A[2] = 18 > 14 // true i.e. swap the both element positions.

A[0] > A[1] = 11 > 14 // false

A[2] > A[3] = 18 > 30 // false A[3] > A[4] = 30 > 19 // true i.e. swap the both element positions. A[2] > A[3] = 18 > 19 // false

A[4] > A[5] = 30 > 37 // false A[5] > A[6] = 37 > 23 // true i.e. swap the both element positions. A[4] > A[5] = 30 > 23 // true i.e. swap the both element positions. A[3] > A[4] = 19 > 23 // false

A[6] > A[7] = 37 > 40 // false A[7] > A[8] = 40 > 29 // true i.e. swap the both element positions. A[6] > A[7] = 37 > 29 // true i.e. swap the both element positions. A[5] > A[6] = 30 > 29 // true i.e. swap the both element positions. A[4] > A[5] = 23 > 29 // false

Pass 3 is complete, and array is fully sorted.

Shell sort program in C language:

#include <stdio.h>
   // Shell sort void shellSort(int array[], int n) {   
  // Rearrange elements at each n/2, n/4, n/8, ... intervals 
  for (int gap = n / 2; gap > 0; gap /= 2) { 
    for (int i = gap; i < n; i += 1) {  
     int temp = array[i];  
     int j;     
  for (j = i; j >= gap && array[j - gap] > temp; j -= gap) { 
        array[j] = array[j - gap];   
    }     
  array[j] = temp;   
  } 
  }
 } 
  // Print an array 
void printArray(int array[], int size) {
   for (int i = 0; i < size; ++i) { 
    printf("%d  ", array[i]);   
} 
  printf("\n"); 
}   // Driver code 
int main() 
{  
 int data[] = {9, 8, 3, 7, 5, 6, 4, 1}; 
  int size = sizeof(data) / sizeof(data[0]);   
shellSort(data, size); 
  printf("Sorted array: \n"); 
  printArray(data, size); 
}