Cycle Sort - GeeksforGeeks

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[] = {4, 5, 2, 1, 5}

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

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.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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
while (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 

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)

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

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

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