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# Cycle Sort

Cycle sort is an in-place, unstable sorting algorithm that is particularly useful when sorting arrays containing elements with a small range of values. It was developed by W. D. Jones and published in 1963.

The basic idea behind cycle sort is to divide the input array into cycles, where each cycle consists of elements that belong to the same position in the sorted output array. The algorithm then performs a series of swaps to place each element in its correct position within its cycle, until all cycles are complete and the array is sorted.

### Here’s a step-by-step explanation of the cycle sort algorithm:

2. Initialize a variable, cycleStart, to 0.
3. For each element in the array, compare it with every other element to its right. If there are any elements that are smaller than the current element, increment cycleStart.
4. If cycleStart is still 0 after comparing the first element with all other elements, move to the next element and repeat step 3.
5. Once a smaller element is found, swap the current element with the first element in its cycle. The cycle is then continued until the current element returns to its original position.

Repeat steps 3-5 until all cycles have been completed.

The array is now sorted.

One of the advantages of cycle sort is that it has a low memory footprint, as it sorts the array in-place and does not require additional memory for temporary variables or buffers. However, it can be slow in certain situations, particularly when the input array has a large range of values. Nonetheless, cycle sort remains a useful sorting algorithm in certain contexts, such as when sorting small arrays with limited value ranges.

Cycle sort is an in-place sorting Algorithm, unstable sorting algorithm, and a comparison sort that is theoretically optimal in terms of the total number of writes to the original array.

• It is optimal in terms of the number of memory writes. It minimizes the number of memory writes to sort (Each value is either written zero times if it’s already in its correct position or written one time to its correct position.)
• It is based on the idea that the array to be sorted can be divided into cycles. Cycles can be visualized as a graph. We have n nodes and an edge directed from node i to node j if the element at i-th index must be present at j-th index in the sorted array.
Cycle in arr[] = {2, 4, 5, 1, 3} Cycle in arr[] = {2, 4, 5, 1, 3}

• Cycle in arr[] = {4, 3, 2, 1} Cycle in arr[] = {4, 3, 2, 1}

We one by one consider all cycles. We first consider the cycle that includes the first element. We find the correct position of the first element, and place it at its correct position, say j. We consider the old value of arr[j] and find its correct position, we keep doing this till all elements of the current cycle are placed at the correct position, i.e., we don’t come back to the cycle starting point.

Pseudocode :

```Begin
for
start:= 0 to n - 2 do
key := array[start]
location := start
for i:= start + 1 to n-1 do
if array[i] < key then
location: =location +1
done
if location = start then
ignore lower part, go for next iteration
while key = array[location] do
location: = location + 1
done
if location != start then
swap array[location] with key
while location != start do
location start

for i:= start + 1 to n-1 do
if array[i] < key then
location: =location +1
done
while key= array[location]
location := location +1
if key != array[location]
Swap array[location] and key
done
done
End```

Explanation :

``` arr[] = {10, 5, 2, 3}
index =  0   1   2   3
cycle_start = 0
item = 10 = arr

Find position where we put the item
pos = cycle_start
i=pos+1
while(i<n)
if (arr[i] < item)
pos++;

We put 10 at arr and change item to
old value of arr.
arr[] = {10, 5, 2, 10}
item = 3

Find position where we put the item = 3
we swap item with element at arr now
arr[] = {10, 3, 2, 10}
item = 5

Again rotate rest cycle that start with index '0' and item = 5
we swap item with element at arr.
arr[] = {10, 3, 5, 10 }
item = 2

Again rotate rest cycle that start with index '0' and item = 2
arr[] = {2, 3,  5, 10}

Above is one iteration for cycle_stat = 0.
Repeat above steps for cycle_start = 1, 2, ..n-2```

Below is the implementation of the above approach:

## CPP

 `// C++ program to implement cycle sort``#include ``using` `namespace` `std;` `// Function sort the array using Cycle sort``void` `cycleSort(``int` `arr[], ``int` `n)``{``    ``// count number of memory writes``    ``int` `writes = 0;` `    ``// traverse array elements and put it to on``    ``// the right place``    ``for` `(``int` `cycle_start = 0; cycle_start <= n - 2; cycle_start++) {``        ``// initialize item as starting point``        ``int` `item = arr[cycle_start];` `        ``// Find position where we put the item. We basically``        ``// count all smaller elements on right side of item.``        ``int` `pos = cycle_start;``        ``for` `(``int` `i = cycle_start + 1; i < n; i++)``            ``if` `(arr[i] < item)``                ``pos++;` `        ``// If item is already in correct position``        ``if` `(pos == cycle_start)``            ``continue``;` `        ``// ignore all duplicate  elements``        ``while` `(item == arr[pos])``            ``pos += 1;` `        ``// put the item to it's right position``        ``if` `(pos != cycle_start) {``            ``swap(item, arr[pos]);``            ``writes++;``        ``}` `        ``// Rotate rest of the cycle``        ``while` `(pos != cycle_start) {``            ``pos = cycle_start;` `            ``// Find position where we put the element``            ``for` `(``int` `i = cycle_start + 1; i < n; i++)``                ``if` `(arr[i] < item)``                    ``pos += 1;` `            ``// ignore all duplicate  elements``            ``while` `(item == arr[pos])``                ``pos += 1;` `            ``// put the item to it's right position``            ``if` `(item != arr[pos]) {``                ``swap(item, arr[pos]);``                ``writes++;``            ``}``        ``}``    ``}` `    ``// Number of memory writes or swaps``    ``// cout << writes << endl ;``}` `// Driver program to test above function``int` `main()``{``    ``int` `arr[] = { 1, 8, 3, 9, 10, 10, 2, 4 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);``    ``cycleSort(arr, n);` `    ``cout << ``"After sort : "` `<< endl;``    ``for` `(``int` `i = 0; i < n; i++)``        ``cout << arr[i] << ``" "``;``    ``return` `0;``}`

## Java

 `// Java program to implement cycle sort` `import` `java.util.*;``import` `java.lang.*;` `class` `GFG {``    ``// Function sort the array using Cycle sort``    ``public` `static` `void` `cycleSort(``int` `arr[], ``int` `n)``    ``{``        ``// count number of memory writes``        ``int` `writes = ``0``;` `        ``// traverse array elements and put it to on``        ``// the right place``        ``for` `(``int` `cycle_start = ``0``; cycle_start <= n - ``2``; cycle_start++) {``            ``// initialize item as starting point``            ``int` `item = arr[cycle_start];` `            ``// Find position where we put the item. We basically``            ``// count all smaller elements on right side of item.``            ``int` `pos = cycle_start;``            ``for` `(``int` `i = cycle_start + ``1``; i < n; i++)``                ``if` `(arr[i] < item)``                    ``pos++;` `            ``// If item is already in correct position``            ``if` `(pos == cycle_start)``                ``continue``;` `            ``// ignore all duplicate elements``            ``while` `(item == arr[pos])``                ``pos += ``1``;` `            ``// put the item to it's right position``            ``if` `(pos != cycle_start) {``                ``int` `temp = item;``                ``item = arr[pos];``                ``arr[pos] = temp;``                ``writes++;``            ``}` `            ``// Rotate rest of the cycle``            ``while` `(pos != cycle_start) {``                ``pos = cycle_start;` `                ``// Find position where we put the element``                ``for` `(``int` `i = cycle_start + ``1``; i < n; i++)``                    ``if` `(arr[i] < item)``                        ``pos += ``1``;` `                ``// ignore all duplicate elements``                ``while` `(item == arr[pos])``                    ``pos += ``1``;` `                ``// put the item to it's right position``                ``if` `(item != arr[pos]) {``                    ``int` `temp = item;``                    ``item = arr[pos];``                    ``arr[pos] = temp;``                    ``writes++;``                ``}``            ``}``        ``}``    ``}` `    ``// Driver program to test above function``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `arr[] = { ``1``, ``8``, ``3``, ``9``, ``10``, ``10``, ``2``, ``4` `};``        ``int` `n = arr.length;``        ``cycleSort(arr, n);` `        ``System.out.println(``"After sort : "``);``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``System.out.print(arr[i] + ``" "``);``    ``}``}` `// Code Contributed by Mohit Gupta_OMG <(0_o)>`

## Python3

 `# Python program to implement cycle sort` `def` `cycleSort(array):``  ``writes ``=` `0``  ` `  ``# Loop through the array to find cycles to rotate.``  ``for` `cycleStart ``in` `range``(``0``, ``len``(array) ``-` `1``):``    ``item ``=` `array[cycleStart]``    ` `    ``# Find where to put the item.``    ``pos ``=` `cycleStart``    ``for` `i ``in` `range``(cycleStart ``+` `1``, ``len``(array)):``      ``if` `array[i] < item:``        ``pos ``+``=` `1``    ` `    ``# If the item is already there, this is not a cycle.``    ``if` `pos ``=``=` `cycleStart:``      ``continue``    ` `    ``# Otherwise, put the item there or right after any duplicates.``    ``while` `item ``=``=` `array[pos]:``      ``pos ``+``=` `1``    ``array[pos], item ``=` `item, array[pos]``    ``writes ``+``=` `1``    ` `    ``# Rotate the rest of the cycle.``    ``while` `pos !``=` `cycleStart:``      ` `      ``# Find where to put the item.``      ``pos ``=` `cycleStart``      ``for` `i ``in` `range``(cycleStart ``+` `1``, ``len``(array)):``        ``if` `array[i] < item:``          ``pos ``+``=` `1``      ` `      ``# Put the item there or right after any duplicates.``      ``while` `item ``=``=` `array[pos]:``        ``pos ``+``=` `1``      ``array[pos], item ``=` `item, array[pos]``      ``writes ``+``=` `1``  ` `  ``return` `writes``  ` `# driver code``arr ``=` `[``1``, ``8``, ``3``, ``9``, ``10``, ``10``, ``2``, ``4` `]``n ``=` `len``(arr)``cycleSort(arr)` `print``(``"After sort : "``)``for` `i ``in` `range``(``0``, n) :``    ``print``(arr[i], end ``=` `' '``)` `# Code Contributed by Mohit Gupta_OMG <(0_o)>`

## C#

 `// C# program to implement cycle sort``using` `System;` `class` `GFG {``    ` `    ``// Function sort the array using Cycle sort``    ``public` `static` `void` `cycleSort(``int``[] arr, ``int` `n)``    ``{``        ``// count number of memory writes``        ``int` `writes = 0;` `        ``// traverse array elements and``        ``// put it to on the right place``        ``for` `(``int` `cycle_start = 0; cycle_start <= n - 2; cycle_start++)``        ``{``            ``// initialize item as starting point``            ``int` `item = arr[cycle_start];` `            ``// Find position where we put the item.``            ``// We basically count all smaller elements``            ``// on right side of item.``            ``int` `pos = cycle_start;``            ``for` `(``int` `i = cycle_start + 1; i < n; i++)``                ``if` `(arr[i] < item)``                    ``pos++;` `            ``// If item is already in correct position``            ``if` `(pos == cycle_start)``                ``continue``;` `            ``// ignore all duplicate elements``            ``while` `(item == arr[pos])``                ``pos += 1;` `            ``// put the item to it's right position``            ``if` `(pos != cycle_start) {``                ``int` `temp = item;``                ``item = arr[pos];``                ``arr[pos] = temp;``                ``writes++;``            ``}` `            ``// Rotate rest of the cycle``            ``while` `(pos != cycle_start) {``                ``pos = cycle_start;` `                ``// Find position where we put the element``                ``for` `(``int` `i = cycle_start + 1; i < n; i++)``                    ``if` `(arr[i] < item)``                        ``pos += 1;` `                ``// ignore all duplicate elements``                ``while` `(item == arr[pos])``                    ``pos += 1;` `                ``// put the item to it's right position``                ``if` `(item != arr[pos]) {``                    ``int` `temp = item;``                    ``item = arr[pos];``                    ``arr[pos] = temp;``                    ``writes++;``                ``}``            ``}``        ``}``    ``}` `    ``// Driver program to test above function``    ``public` `static` `void` `Main()``    ``{``        ``int``[] arr = { 1, 8, 3, 9, 10, 10, 2, 4 };``        ``int` `n = arr.Length;``        ` `        ``// Function calling``        ``cycleSort(arr, n);` `        ``Console.WriteLine(``"After sort : "``);``        ``for` `(``int` `i = 0; i < n; i++)``            ``Console.Write(arr[i] + ``" "``);``    ``}``}` `// This code is contributed by Nitin Mittal`

## Javascript

 ``

Output

```After sort :
1 2 3 4 8 9 10 10 ```

Time Complexity Analysis

• Worst Case: O(n2
• Average Case: O(n2
• Best Case: O(n2)

Auxiliary Space: O(1)

• The space complexity is constant cause this algorithm is in place so it does not use any extra memory to sort.

Method 2: This method is only applicable when given array values or elements are in the range of 1 to N or  0 to N. In this method, we do not need to rotate an array

Approach : All the given array values should be in the range of 1 to N or 0 to N. If the range is 1 to N  then every array element’s correct position will be the index == value-1 i.e. means at the 0th index value will be 1 similarly at the 1st index position value will be 2 and so on till nth value.

similarly for 0 to N values correct index position of each array element or value will be the same as its value i.e. at 0th index 0 will be there 1st position 1 will be there.

Explanation :

```arr[] = {5, 3, 1, 4, 2}
index =  0  1  2  3  4

i  = 0;
while( i < arr.length)
correctposition = arr[i]-1;

find ith item correct position
for the first time i = 0 arr = 5 correct index of 5 is 4 so arr[i] - 1 = 5-1 = 4

if( arr[i] <= arr.length && arr[i] != arr[correctposition])

arr[i] = 5 and arr[correctposition] = 4
so 5 <= 5 && 5 != 4 if condition true
now swap the 5 with 4

int temp = arr[i];
arr[i] = arr[correctposition];
arr[correctposition] = temp;

now resultant arr at this after 1st swap
arr[] = {2, 3, 1, 4, 5} now 5 is shifted at its correct position

now loop will run again check for i = 0 now arr[i] is = 2
after swapping 2 at its correct position
arr[] = {3, 2, 1, 4, 5}

now loop will run again check for i = 0 now arr[i] is = 3
after swapping 3 at its correct position
arr[] = {1, 2, 3, 4, 5}

now loop will run again check for i = 0 now arr[i] is = 1
this time  1 is at its correct position so else block will execute and i will increment i = 1;
once i exceeds the size of array will get array sorted.
arr[] = {1, 2, 3, 4, 5}

else

i++;
loop end;

once while loop end we get sorted array just print it
for( index = 0 ; index < arr.length; index++)
print(arr[index] + " ")
sorted arr[] = {1, 2, 3, 4, 5}```

Below is the implementation of the above approach:

## C++

 `#include ``using` `namespace` `std;` `void` `cyclicSort(``int` `arr[], ``int` `n){``  ``int` `i = 0;``  ``while``(i < n)``  ``{``    ``// as array is of 1 based indexing so the``    ``// correct position or index number of each``    ``// element is element-1 i.e. 1 will be at 0th``    ``// index similarly 2 correct index will 1 so``    ``// on...``    ``int` `correct = arr[i] - 1 ;``    ``if``(arr[i] != arr[correct]){` `      ``// if array element should be lesser than``      ``// size and array element should not be at``      ``// its correct position then only swap with``      ``// its correct position or index value``      ``swap(arr[i], arr[correct]) ;``    ``}``else``{` `      ``// if element is at its correct position``      ``// just increment i and check for remaining``      ``// array elements``      ``i++ ;``    ``}``  ``}` `}` `void` `printArray(``int` `arr[], ``int` `size)``{``  ``int` `i;``  ``for` `(i = 0; i < size; i++)``    ``cout << arr[i] << ``" "``;``  ``cout << endl;``}` `int` `main() {` `  ``int` `arr[] = { 3, 2, 4, 5, 1};``  ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);``  ``cout << ``"Before sorting array: \n"``;``  ``printArray(arr, n);``  ``cyclicSort(arr, n);``  ``cout << ``"Sorted array: \n"``;``  ``printArray(arr, n);``  ``return` `0;` `}`

## Java

 `// java program to check implement cycle sort``import` `java.util.*;``public` `class` `MissingNumber {``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int``[] arr = { ``3``, ``2``, ``4``, ``5``, ``1` `};``        ``int` `n = arr.length;``        ``System.out.println(``"Before sort :"``);``        ``System.out.println(Arrays.toString(arr));``        ``CycleSort(arr, n);``        ` `    ``}` `    ``static` `void` `CycleSort(``int``[] arr, ``int` `n)``    ``{``        ``int` `i = ``0``;``        ``while` `(i < n) {``            ``// as array is of 1 based indexing so the``            ``// correct position or index number of each``            ``// element is element-1 i.e. 1 will be at 0th``            ``// index similarly 2 correct index will 1 so``            ``// on...``            ``int` `correctpos = arr[i] - ``1``;``            ``if` `(arr[i] < n && arr[i] != arr[correctpos]) {``                ``// if array element should be lesser than``                ``// size and array element should not be at``                ``// its correct position then only swap with``                ``// its correct position or index value``                ``swap(arr, i, correctpos);``            ``}``            ``else` `{``                ``// if element is at its correct position``                ``// just increment i and check for remaining``                ``// array elements``                ``i++;``            ``}``        ``}``            ``System.out.println(``"After sort :  "``);``            ``System.out.print(Arrays.toString(arr));``        ` `        ` `    ``}` `    ``static` `void` `swap(``int``[] arr, ``int` `i, ``int` `correctpos)``    ``{``    ``// swap elements with their correct indexes``        ``int` `temp = arr[i];``        ``arr[i] = arr[correctpos];``        ``arr[correctpos] = temp;``    ``}``}``// this code is contributed by devendra solunke`

## Python3

 `# Python program to check implement cycle sort``def` `cyclicSort(arr, n):``    ``i ``=` `0``    ``while` `i < n:``          ``# as array is of 1 based indexing so the``        ``# correct position or index number of each``        ``# element is element-1 i.e. 1 will be at 0th``        ``# index similarly 2 correct index will 1 so``        ``# on...``        ``correct ``=` `arr[i] ``-` `1``        ``if` `arr[i] !``=` `arr[correct]:``          ` `              ``# if array element should be lesser than``              ``# size and array element should not be at``              ``# its correct position then only swap with``              ``# its correct position or index value``            ``arr[i], arr[correct] ``=` `arr[correct], arr[i]``        ``else``:``              ``# if element is at its correct position``              ``# just increment i and check for remaining``              ``# array elements``            ``i ``+``=` `1`  `def` `printArray(arr):``    ``print``(``*``arr)` `arr ``=` `[``3``, ``2``, ``4``, ``5``, ``1``]``n ``=` `len``(arr)``print``(``"Before sorting array:"``)``printArray(arr)` `# Function Call``cyclicSort(arr, n)``print``(``"Sorted array:"``)``printArray(arr)` `# This Code is Contributed by Prasad Kandekar(prasad264)`

## C#

 `using` `System;` `public` `class` `GFG {` `    ``static` `void` `CycleSort(``int``[] arr, ``int` `n)``    ``{``        ``int` `i = 0;``        ``while` `(i < n) {``            ``// as array is of 1 based indexing so the``            ``// correct position or index number of each``            ``// element is element-1 i.e. 1 will be at 0th``            ``// index similarly 2 correct index will 1 so``            ``// on...``            ``int` `correctpos = arr[i] - 1;``            ``if` `(arr[i] < n && arr[i] != arr[correctpos]) {``                ``// if array element should be lesser than``                ``// size and array element should not be at``                ``// its correct position then only swap with``                ``// its correct position or index value``                ``swap(arr, i, correctpos);``            ``}``            ``else` `{``                ``// if element is at its correct position``                ``// just increment i and check for remaining``                ``// array elements``                ``i++;``            ``}``        ``}``        ``Console.Write(``"\nAfter sort : "``);``        ``for` `(``int` `index = 0; index < n; index++)``            ``Console.Write(arr[index] + ``" "``);``    ``}` `    ``static` `void` `swap(``int``[] arr, ``int` `i, ``int` `correctpos)``    ``{``        ``// swap elements with their correct indexes``        ``int` `temp = arr[i];``        ``arr[i] = arr[correctpos];``        ``arr[correctpos] = temp;``    ``}` `    ``static` `public` `void` `Main()``    ``{``        ``// Code``        ``int``[] arr = { 3, 2, 4, 5, 1 };``        ``int` `n = arr.Length;``        ``Console.Write(``"Before sort : "``);``        ``for` `(``int` `i = 0; i < n; i++)``            ``Console.Write(arr[i] + ``" "``);``        ``CycleSort(arr, n);``    ``}``}``// This code is contributed by devendra solunke`

## Javascript

 `// JavaScript code for the above code``function` `cyclicSort(arr, n) {``    ``var` `i = 0;``    ``while` `(i < n)``    ``{``    ` `        ``// as array is of 1 based indexing so the``        ``// correct position or index number of each``        ``// element is element-1 i.e. 1 will be at 0th``        ``// index similarly 2 correct index will 1 so``        ``// on...``        ``let correct = arr[i] - 1;``        ``if` `(arr[i] !== arr[correct])``        ``{``        ` `            ``// if array element should be lesser than``            ``// size and array element should not be at``            ``// its correct position then only swap with``            ``// its correct position or index value``            ``[arr[i], arr[correct]] = [arr[correct], arr[i]];``        ``}``        ``else` `{``            ``// if element is at its correct position``            ``// just increment i and check for remaining``            ``// array elements``            ``i++;``        ``}``    ``}``}` `function` `printArray(arr, size) {``    ``for` `(``var` `i = 0; i < size; i++) {``        ``console.log(arr[i] + ``" "``);``    ``}``    ``console.log(``"\n"``);``}` `var` `arr = [3, 2, 4, 5, 1];``var` `n = arr.length;``console.log(``"Before sorting array: \n"``);``printArray(arr, n);``cyclicSort(arr, n);``console.log(``"Sorted array: \n"``);``printArray(arr, n);` `// This Code is Contributed by Prasad Kandekar(prasad264)`

Output

```Before sorting array:
3 2 4 5 1
Sorted array:
1 2 3 4 5 ```

Time Complexity Analysis:

• Worst Case : O(n)
• Average Case: O(n)
• Best Case : O(n)

Auxiliary Space: O(1)

1. No additional storage is required.
2.  in-place sorting algorithm.
3.  A minimum number of writes to the memory
4.  Cycle sort is useful when the array is stored in EEPROM or FLASH.

1.  It is not mostly used.
2.  It has more time complexity o(n^2)
3.  Unstable sorting algorithm.

## Application  of Cycle sort:

• This sorting algorithm is best suited for situations where memory write or swap operations are costly.
• Useful for complex problems.

Reference:
https://en.wikipedia.org/wiki/Cycle_sort
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