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Cycle Sort

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  • Difficulty Level : Medium
  • Last Updated : 31 Aug, 2022
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Cycle sort is an in-place sorting Algorithm, unstable sorting algorithm, 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 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 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-sort

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

cyclc-sort2

We one by one consider all cycles. We first consider the cycle that includes first element. We find correct position of first element, place it at its correct position, say j. We consider old value of arr[j] and find its correct position, we keep doing this till all elements of current cycle are placed at correct position, i.e., we don’t come back to 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
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[0]

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

Again rotate rest cycle that start with index '0' 
Find position where we put the item = 3 
we swap item with element at arr[1] 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[2].
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

 

CPP




// C++ program to implement cycle sort
#include <iostream>
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[0]);
    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.Write("After sort : ");
        for (int i = 0; i < n; i++)
            Console.Write(arr[i] + " ");
    }
}
 
// This code is contributed by Nitin Mittal

Javascript




<script>
// Javascript program to implement cycle sort
 
    // Function sort the array using Cycle sort
    function cycleSort(arr, n)
    {
     
        // count number of memory writes
        let writes = 0;
   
        // traverse array elements and put it to on
        // the right place
        for (let cycle_start = 0; cycle_start <= n - 2; cycle_start++)
        {
         
            // initialize item as starting point
            let item = arr[cycle_start];
   
            // Find position where we put the item. We basically
            // count all smaller elements on right side of item.
            let pos = cycle_start;
            for (let 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)
            {
                let 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 (let 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]) {
                    let temp = item;
                    item = arr[pos];
                    arr[pos] = temp;
                    writes++;
                }
            }
        }
    }
      
// Driver code   
 
    let arr = [ 1, 8, 3, 9, 10, 10, 2, 4 ];
       let n = arr.length;
       cycleSort(arr, n);
  
      document.write("After sort : " + "<br/>");
       for (let i = 0; i < n; i++)
           document.write(arr[i] + " ");
   
  // This code is contributed by susmitakundugoaldanga.
</script>

Output: 
 

After sort : 
1 2 3 4 8 9 10 10 

Time Complexity : O(n2

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

Space complexity :

  • The space complexity is constant cause this algorithm is in place so it does not use any extra memory to sort.
  • Auxiliary space: o(1)

Method 2 : This methods is only applicable when given array values or elements are in range of 1 to N or  0 to N. In this method we do not need to rotate 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 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 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 postioin 
     for the first time i = 0 arr[0] = 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 swaping 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 swaping 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 <iostream>
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[0]);
  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

C#




using System;
 
public class GFG {
 
    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);
    }
 
    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("After sort : ");
        for (int index = 0; index < n; i++)
            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;
    }
}
// this code is contributed by devendra solunke

Output : 

before sorting :
[3, 2, 4, 5, 1]
after cycle sort : 
[1, 2, 3, 4, 5]

Time Complexity : O(n)

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

Space complexity :

The space complexity is constant cause this algorithm is in place so it does not use any extra memory to sort.
Auxiliary space: o(1)

Advantage :

  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. 

Disadvantage:

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

Application:

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

https://youtu.be/gZNOM_yMdSQ

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Reference: 
https://en.wikipedia.org/wiki/Cycle_sort
This article is contributed by Nishant Singh. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 


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