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Shell Sort:Shellsort, also known as Shell sort or Shell's method, is an in-placecomparison sort. This techniqueis also called sorting with diminishing increments It generalizes an exchanging sort, such asinsertion orbubble sort, by starting the comparison and exchange of elements with elements that are farapart before finishing with neighbouring elements. Starting with far apart elements can move some out-of-place elements into position faster than a simple nearest neighbour exchange.Donald Shell

published the first version of this sort in 1959. The running time of Shell sort is heavily dependent on thegap sequence it uses. For many practical variants, determining theirtime complexity remains an openproblem.

Algorithm:

1.Establish an array a[] of n elements.2.set increment inc to n/2.3.divide the array into inc sub arrays with inc as the increment size.4.sort these sub arrays individually.5.Reduce inc to inc/2.6.Repeat steps 3 through 5 tll inc is equal to 1.7.Return the sorted array.

C++ implementation of Shell Sort:

#includeu sin g n amesp ace std;

in t shell(int x[], int n)

{int i,j, k, no, inc;

fo r (inc = n/2; inc >=1; inc/=2)

for (k = 0; k < inc ; k++)

{/* This is insertion sort */

fo r (i = k; i < n-inc; i+=inc)

{ no = x[i+inc];

fo r (j = i;j >= k;j-=inc)

if(no > x[j])

break ;

else x[j+inc] = x[j];

x[j+inc] = no;

}

/* insertion sort ends */

}

retu rn 0;

}

int main()

{ int n,a[100];

coutn;

cout

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Ou pu

Java mp emen a on

import ava ut Scanner

p u b lic class sort {

p u b lic static v oid shell (in t x[], int n)

{in t i,j, k, no, inc;

fo r (inc = n/2; inc >=1; inc/=2)

for (k = 0; k < inc ; k++)

{

for (i = k; i < n-inc; i+=inc)

{ no = x[i+inc];

fo r (j = i;j >= k;j-=inc)

if(no > x[j])

break ;

else x[j+inc] = x[j];

x[j+inc] = no;

}}

}

p u b lic static v oid main(String[] args){

in t n;

in t a[] = n ew int[100];

System.out.println("enter no of elements:");

Scanner sc = n ew Scanner(System.in);

n = sc.nextInt();

System.out.println("enter elements:");

for(int i=0;i

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Performance of algorithm:

The analysis of shell sort requires a lot of mathematical analysis.Eventhough at the core of the shell sort

,we have insertion sort,the analysis varies from that of insertion sort,The efficiency of the sorts depends in

the values of the diminishing increments,They should have prime numbers so as to effect proper mixing

ofelements.If that is achieved for the last pass,which is like implementing classical insertion sort,minimum

numbers of shifting are required.mathematical analysis shows shell sort to be more efficient than classical

insertion sort with O(n1.5).

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Merge Sort:

A Merge sort (also commonly spelled mergesort) is anO(n log n)comparison-based sorting algorithm.

Most implementations produce astable sort, which means that the implementation preserves the input

order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented

byJohn von Neumann in 1945.

Algorithm:

1.Establish an array a[] of n elements.

2.Consider size equal to unity.

3.Subdivide the array in sub arrays of size n.

4.Merge the sub arrays in pairs of 2 such that they remain sorted.

5.Double the size.

6. repeat steps 3 to 5 till size exceeds array size.

7. return sorted array.

C++ Implementation .#include

u sin g n amesp ace std;

int mergesort(int x[], in t lb, int ub){ int merge(int [], int, int, int);

int mid;

if(lb < ub)

{mid = (lb + ub) / 2;

mergesort(x, lb, mid);

mergesort(x, mid+1, ub);

merge(x, lb, mid, ub);

}

return 0;

}

int merge(int x[], int lb1, int ub1, int ub2)

{

int temp[10], i = lb1,j = ub1 + 1, k = 0;

while(i

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Ou pu

Java mp emen a onimport ava ut Scannerp u

b lic class sort {

p u b lic static v oid mergesort(int x[], in t lb, int ub){in t mid;

if(lb < ub)

{mid = (lb + ub) / 2;

mergesort(x, lb, mid);

mergesort(x, mid+1, ub);

merge(x, lb, mid, ub);

}

}

priv ate static v oid merge(int x[], in t lb1, int ub1, int ub2){

int temp[], i = lb1,j = ub1 + 1, k = 0;

temp = n ew in t[10];

while(i

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Ou pu

Pe o mance o a go hm

Suppose ha he e s ze s o he powe 2 say n=2^m so ha m= ogn o he base 2 n Me ge so s

t t:

rf r f l rit :

t t t fil i i f t r , , t t l t t .I r rt it iobvious that no longer than m passes will be required. During each pass, maximum n comparisons will be

required. This indicates that maximum m*n comparisons will be required which implies that nlogn.Thus

merge sort is O(nlogn).

You can observe the effective change in runtime (exeution time).

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Quick Sort:

Quicksort, or partition-exchange sort, is asorting algorithm developed byTony Hoare that,on average,

makes O(n log n) comparisons to sort n items. In theworst case, it makes O(n2) comparisons, though this

behavior is rare. Quicksort is often faster in practice than other O(n log n) algorithms. Additionally,quicksort's sequential and localized memory references work well with acache. Quicksort is acomparisonsort and, in efficient implementations, is not astable sort. Quicksort can be implemented with anin-place

partitioning algorithm,so the entire sort can be done with only O(log n) additional space used by the stackduring the recursion.

Algorithm:

1. Partition the current array into two sub arrays.2. Invoke quicksort to left sub array.3. Invoke quicksort to right sub array.

C++ Implementation:#include


int partition (int x[], in t lb, int ub)

{int pos = lb, i, temp;

for (i = lb+1; i

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Output

Java mp emen a on



priv ate static in t partition (int x[], int lb, in t ub)

{in t pos = lb, i, temp;

fo r (i = lb+1; i

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Ou pu

Pe o mance o a go hm

Quicksort takes O n log n time on average, when the input is a random permutation. Why? For a start, it is nothard to see that the partition operation takes O n time.In the most unbalanced case, each time we perform apartition we divide the list into two sublists of size 0 and n-1 (for example, if all elements of the array are equal).

t t:

rf r f l rit :

( )( )

This means each recursive call processes a list of size one less than the previous list. Consequently, we canmake n-1 nested calls before we reach a list of size 1. This means that thecalltree is a linear chain of n-1 nestedcalls. The ith call does O(n-1) work to do the partition, and O(n^2) , so in that case Quicksorttakes O(n^2) time.That is the worst case: given knowledge of which comparisons are performed by the sort, thereare adaptive algorithms that are effective at generating worst-case input for quicksort on-the-fly, regardless of thepivot selection strategy
http://en.wikipedia.org/wiki/Call_stackhttp://en.wikipedia.org/wiki/Call_stackhttp://en.wikipedia.org/wiki/Call_stackhttp://en.wikipedia.org/wiki/Call_stackhttp://en.wikipedia.org/wiki/Call_stackhttp://en.wikipedia.org/wiki/Call_stack

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Radix Sort:

radix sort is a non-comparative integer sortingalgorithm that sorts data with integer keys by grouping keys bythe individual digits which share the samesignificant position and value. Apositionalnotation is required, butbecause integers can represent strings of characters (e.g., names or dates) and specially formatted floating pointnumbers,radix sort is not limited to integers. Radix sort dates back as far as 1887 to the work of HermanHollerith ontabulatingmachines.

Most digital computers internally represent all of their data as electronic representations of binary numbers, soprocessing the digits of integer representations by groups of binary digit representations is most convenient. Twoclassifications of radix sorts areleastsignificantdigit (LSD) radix sorts andmostsignificantdigit (MSD) radixsorts. LSD radix sorts process the integer representations starting from the least digit and move towards the mostsignificant digit. MSD radix sorts work the other way around.

The integer representations that are processed by sorting algorithms are often called "keys", which can exist allby themselves or be associated with other data. LSD radix sorts typically use the following sorting order: shortkeys come before longer keys, and keys of the same length are sorted lexicographically. This coincides with thenormal order of integer representations, such as the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. MSD radix sorts uselexicographic order, which is suitable for sorting strings, such as words, or fixed-length integer representations. Asequence such as "b, c, d, e, f, g, h, i, j, ba" would be lexicographically sorted as "b, ba, c, d, e, f, g, h, i, j". Iflexicographic ordering is used to sort variable-length integer representations, then the representations of thenumbers from 1 to 10 would be output as 1, 10, 2, 3, 4, 5, 6, 7, 8, 9, as if the shorter keys were left-justified andpadded on the right with blank characters to make the shorter keys as long as the longest key for the purpose ofdetermining sorted order.

Algorithm:

1. Establish array a[] of n elements.

2. Call bucket sort repeatedly on least to most significant digit of each element as the key.

3. Return the stored array.

C++ Implementation:

#include

#include


ty ped ef in t qelement;

ty ped ef stru ct node{ qelement data;

stru ct node *next;

} node;

ty ped ef stru ct

{ node *first, *last;}qtype;

v oid initialise(qtype *qptr)

{qptr->first = NULL;}

int addq(qtype *qptr, qelement no)

{node *p;

p = (node*) malloc (sizeo f(node));

if(p==NULL) return 0;

p->data = no;

p->next = NULL;

if(qptr->first == NULL)qptr->first = p;

else qptr->last->next = p;

qptr->last = p;

return 1;

}

int delq (qtype *qptr, qelement *noptr)

{node *p = qptr->first;

if (p== NULL)return 0;

*noptr = p->data;

qptr->first = p->next;

free (p);

return 1;

}

int radix (int x[], int n, int digs)

{qtype q[10];
http://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Herman_Hollerithhttp://en.wikipedia.org/wiki/Herman_Hollerithhttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Most_significant_digithttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Least_significant_digithttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Tabulating_machineshttp://en.wikipedia.org/wiki/Herman_Hollerithhttp://en.wikipedia.org/wiki/Herman_Hollerithhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Positional_notationhttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sort

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{ while (delq(&q[i], &no))

x[j++] = no;

}

}

retu rn ;

}

int main()

{ int n,a[100];

coutn;

cout

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for (int i=0; i0) i /= div;

retu rn i;

}

priv ate static int findMaxNumberOfDigits() {

int max = Integ er.MIN_VALUE;

fo r (int e : unsorted) {

Strin g str = Strin g.valueOf(e);

int length = str.length();

if (length>max) max = length;

}

retu rn max;

}

}

Ou pu

Performance of Algorithm:The logarithm takes O(n) time per bucket sort. There are logk=O(logn) bucket sorts.So the total time is

O(nlogk),where k is the largest key in the data.

t t:

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Heap Sort:

Heapsort is acomparison-based sorting algorithm to create asorted array (or list), and is part oftheselection sort family. Although somewhat slower in practice on most machines than a well-implementedquicksort, it has the advantage of a more favourable worst-caseO(n log n) runtime. Heapsortis anin-place algorithm, but it is not astable sort. It was invented by J. W. J. Williams in 1964.

The heapsort algorithm can be divided into two parts.

In the first step, aheap isbuilt out of the data.In the second step, a sorted array is created by repeatedly removing the largest element from the heap, andinserting it into the array. The heap is reconstructed after each removal. Once all objects have been removedfrom the heap, we have a sorted array. The direction of the sorted elements can be varied by choosing a min-heap or max-heap in step one.

Heapsort can be performed in place. The array can be split into two parts, the sorted array and the heap. Thestorage of heaps as arrays is diagrammedhere. The heap's invariant is preserved after each extraction, so theonly cost is that of extraction.

Algorithm:

1. Establish the array a[] of n elements.

2. With the elements of the array create an in place max heap.

3. Swap the first element with the last element of the array.

4. Without considering the last element ,adjust the elements of the array so as to again represent max

heap.5. Repeat steps 3 and 4 for subsequent element.

6. Return the sorted array.

C++ Implementation

#include


v oid swap(in t *a, int *b)

{ int temp = *a;

*a = *b;

*b = temp;

}

v oid heapup(int x[], in t root, int bottom){

int parent, temp;

if(bottom > root)

{ parent = (bottom -1) / 2;

if (x[parent] < x[bottom])

{swap(&x[parent], &x[bottom]);

heapup(x, root,parent);

}

}

}

v oid heapdown(int x[], in t root, in t bottom)

{in t lchild= 2*root+1, rchild = 2*root+2, maxchild, temp;

if (lchild x[rchild])maxchild = lchild;

else maxchild = rchild;

if(x[root] < x[maxchild])

{swap(&x[root], &x[maxchild]);

heapdown(x, maxchild,bottom);

}

}

}

v oid heapsort(int x[], int n)

{int i, temp;

for(i = 0; i < n-1; i++)

heapup(x, 0, i+1);

for(i = n-1; i > 0; i){

swap(&x[0], &x[i]);

heapdown(x, 0, i-1);

}}

int main()
http://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Sorted_arrayhttp://en.wikipedia.org/wiki/Sorted_arrayhttp://en.wikipedia.org/wiki/Sorted_arrayhttp://en.wikipedia.org/wiki/Selection_sorthttp://en.wikipedia.org/wiki/Selection_sorthttp://en.wikipedia.org/wiki/Selection_sorthttp://en.wikipedia.org/wiki/Quicksorthttp://en.wikipedia.org/wiki/Quicksorthttp://en.wikipedia.org/wiki/Quicksorthttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/In-place_algorithmhttp://en.wikipedia.org/wiki/In-place_algorithmhttp://en.wikipedia.org/wiki/In-place_algorithmhttp://en.wikipedia.org/wiki/Stable_sorthttp://en.wikipedia.org/wiki/Stable_sorthttp://en.wikipedia.org/wiki/Stable_sorthttp://en.wikipedia.org/wiki/Heap_(data_structure)http://en.wikipedia.org/wiki/Heap_(data_structure)http://en.wikipedia.org/wiki/Heap_(data_structure)http://en.wikipedia.org/wiki/Binary_heap#Building_a_heaphttp://en.wikipedia.org/wiki/Binary_heap#Building_a_heaphttp://en.wikipedia.org/wiki/Binary_heap#Building_a_heaphttp://en.wikipedia.org/wiki/Binary_heap#Heap_implementationhttp://en.wikipedia.org/wiki/Binary_heap#Heap_implementationhttp://en.wikipedia.org/wiki/Binary_heap#Heap_implementationhttp://en.wikipedia.org/wiki/Binary_heap#Heap_implementationhttp://en.wikipedia.org/wiki/Binary_heap#Heap_implementationhttp://en.wikipedia.org/wiki/Binary_heap#Building_a_heaphttp://en.wikipedia.org/wiki/Binary_heap#Building_a_heaphttp://en.wikipedia.org/wiki/Heap_(data_structure)http://en.wikipedia.org/wiki/Heap_(data_structure)http://en.wikipedia.org/wiki/Stable_sorthttp://en.wikipedia.org/wiki/Stable_sorthttp://en.wikipedia.org/wiki/In-place_algorithmhttp://en.wikipedia.org/wiki/In-place_algorithmhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Quicksorthttp://en.wikipedia.org/wiki/Quicksorthttp://en.wikipedia.org/wiki/Selection_sorthttp://en.wikipedia.org/wiki/Selection_sorthttp://en.wikipedia.org/wiki/Sorted_arrayhttp://en.wikipedia.org/wiki/Sorted_arrayhttp://en.wikipedia.org/wiki/Comparison_sorthttp://en.wikipedia.org/wiki/Comparison_sort

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{ int n,a[100];

coutn;

cout

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for(i = 0; i < n-1; i++)

heapup(x, 0, i+1);

for(i = n-1; i > 0; i--)

{swap(x[0], x[i]);

heapdown(x, 0, i-1);

}

}

p u b lic static void main(Strin g[] args){

int n;

int a[] = n ew int[100]; Sy stem

.out.println("enter no of elements:");

Scanner sc = n ew Scanner(Sy stem.in);

n = sc.nextInt(); Sy stem

.out.println("enter elements:"); fo r

(int i=0;i

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fo r(i=0;i

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Java implementation:



p u b lic static int bucket(int n,int x[]){

int i,j,count[];

count = n ew int[100];

fo r(i=0;i

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if(a[j]

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Java Implementation

imp ort java.util.*;

p u b lic class insertion {

p u b lic int a[]=new int[100];

int max;

p u b lic void sort(int n){

max=n;

int temp;

fo r(int i=0;i0;j--){

if(a[j]

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}

}

}

}

void print(){

fo r(int i=0;i

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Java Implementation


p u b lic class bubble {


int max;


max=n;

int temp;

fo r(int i=0;i

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{min=a[j];

k=j;

}

}

a[k]=a[i];

a[i]=min;

}

}

void print(){

fo r(int i=0;i

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Java Implementation


p u b lic class selection {


int max;


max=n;

int temp,min,k=0;

fo r(int i=0;i

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