Algorithm

Getting Started with AlgorithmWhat is an Algorithm?

Characteristics of Algorithm1 Topic

Analysis Framework

Performance Analysis3 Topics

Mathematical Analysis2 Topics

Sorting AlgorithmSorting Algorithm10 Topics

Searching Algorithm6 Topics

Fundamental of Data StructuresStacks

Queues

Graphs

Trees

Sets

Dictionaries

Divide and ConquerGeneral Method

Binary Search

Recurrence Equation for Divide and Conquer

Finding the Maximum and Minimum

Merge Sort

Quick Sort

Stassen’s Matrix Multiplication

Advantages and Disadvantages of Divide and Conquer

Decrease and ConquerInsertion Sort

Topological Sort

Greedy MethodGeneral Method

Coin Change Problem

Knapsack Problem

Job Sequencing with Deadlines

Minimum Cost Spanning Trees2 Topics

Single Source Shortest Paths1 Topic

Optimal Tree Problem1 Topic

Transform and Conquer Approach1 Topic

Dynamic ProgrammingGeneral Method with Examples

Multistage Graphs

Transitive Closure1 Topic

All Pairs Shortest Paths6 Topics

BacktrackingGeneral Method

NQueens Problem

Sum of Subsets problem

Graph Coloring

Hamiltonian Cycles

Branch and Bound2 Topics

0/1 Knapsack problem2 Topics

NPComplete and NPHard Problems1 Topic
Shell Sort
Shell sort is an algorithm that first sorts the elements far apart from each other and successively reduces the interval between the elements to be sorted. It is a generalized version of insertion sort.
In shell sort, elements at a specific interval are sorted. The interval between the elements is gradually decreased based on the sequence used. the performance of the shell sort depends on the type of sequence used for a given input array.
Some of the optimal sequences used are:
 Shell’s original sequence: N/2 , N/4 , …, 1
 Knuth’s increments: 1, 4, 13, …, (3k – 1) / 2
 Sedgewick’s increments: 1, 8, 23, 77, 281, 1073, 4193, 16577…4j+1+ 3·2j+ 1.
 Hibbard’s increments: 1, 3, 7, 15, 31, 63, 127, 255, 511…
 Papernov & Stasevich increment: 1, 3, 5, 9, 17, 33, 65,…
 Pratt: 1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81….
How Shell Sort Works?
 Suppose, we need to sort the following array.
2.We are using the shell’s original sequence (N/2, N/4, ...1
) as intervals in our algorithm.
In the first loop, if the array size is N = 8
then, the elements lying at the interval of N/2 = 4
are compared and swapped if they are not in order.
 The 0th element is compared with the 4th element.
 If the 0th element is greater than the 4th one then, the 4th element is first stored in
temp
variable and the 0th element (ie. greater element) is stored in the 4th position and the element stored intemp
is stored in the 0th position.
This process goes on for all the remaining elements.
3.In the second loop, an interval of N/4 = 8/4 = 2
is taken and again the elements lying at these intervals are sorted.
You might get confused at this point.
The elements at 4th and 2nd position are compared. The elements at 2nd and 0th position are also compared. All the elements in the array lying at the current interval are compared.
4.The same process goes on for remaining elements.
5. Finally, when the interval is N/8 = 8/8 =1
then the array elements lying at the interval of 1 are sorted. The array is now completely sorted.
Shell Sort Algorithm
shellSort(array, size)
for interval i < size/2n down to 1
for each interval "i" in array
sort all the elements at interval "i"
end shellSort
Shell Sort Program in C
// Shell Sort in C programming
#include <stdio.h>
void shellSort(int array[], int n){
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;
}
}
}
void printArray(int array[], int size){
for(int i=0; i<size; ++i){
printf("%d ", array[i]);
}
printf("\n");
}
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);
}
Complexity
Shell sort is unstable sorting algorithm because this algorithm does not examine the elements lying in between the intervals.
Time Complexity
 Worst Case Complexity: less than or equal to
O(n2)
Worst case complexity for shell sort is always less than or equal toO(n2)
.
According to Poonen Theorem, worst case complexity for shell sort isΘ(NlogN)2/(log logN)2)
orΘ(NlogN)2/log logN)
orΘ(N(logN)2)
or something in between.
 Best Case Complexity:
O(n*log n)
When the array is already sorted, the total number of comparison for each interval (or increment) is equal to the size of the array.
 Average Case Complexity:
O(n*log n)
It is aroundO(n1.25)
.
The complexity depends on the interval chosen. The above complexities differ for different increment sequence chosen. Best increment sequence is unknown.
Space Complexity:
The space complexity for shell sort is O(1)
.
Shell Sort Applications
Shell sort is used when:
 calling a stack is overhead. uClibc library uses this sort.
 recursion exceeds a limit. bzip2 compressor uses it.
 Insertion sort does not perform well when the close elements are far apart. Shell sort helps in reducing the distance between the close elements. Thus, there will be less number of swappings to be performed.