# Sort n numbers in range from 0 to n^2 – 1 in linear time

Given an array of numbers of size n. It is also given that the array elements are in range from 0 to n2 – 1. Sort the given array in linear time.

Examples:

```Since there are 5 elements, the elements can be from 0 to 24.
Input: arr[] = {0, 23, 14, 12, 9}
Output: arr[] = {0, 9, 12, 14, 23}

Since there are 3 elements, the elements can be from 0 to 8.
Input: arr[] = {7, 0, 2}
Output: arr[] = {0, 2, 7}
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

Solution: If we use Counting Sort, it would take O(n^2) time as the given range is of size n^2. Using any comparison based sorting like Merge Sort, Heap Sort, .. etc would take O(nLogn) time.
Now question arises how to do this in 0(n)? Firstly, is it possible? Can we use data given in question? n numbers in range from 0 to n2 – 1?
The idea is to use Radix Sort. Following is standard Radix Sort algorithm.

```1) Do following for each digit i where i varies from least
significant digit to the most significant digit.
…………..a) Sort input array using counting sort (or any stable
sort) according to the i’th digit
```

Let there be d digits in input integers. Radix Sort takes O(d*(n+b)) time where b is the base for representing numbers, for example, for decimal system, b is 10. Since n2-1 is the maximum possible value, the value of d would be O(logb(n)). So overall time complexity is O((n+b)*O(logb(n)). Which looks more than the time complexity of comparison based sorting algorithms for a large k. The idea is to change base b. If we set b as n, the value of O(logb(n)) becomes O(1) and overall time complexity becomes O(n).

```arr[] = {0, 10, 13, 12, 7}

Let us consider the elements in base 5. For example 13 in
base 5 is 23, and 7 in base 5 is 12.
arr[] = {00(0), 20(10), 23(13), 22(12), 12(7)}

After first iteration (Sorting according to the last digit in
base 5),  we get.
arr[] = {00(0), 20(10), 12(7), 22(12), 23(13)}

After second iteration, we get
arr[] = {00(0), 12(7), 20(10), 22(12), 23(13)}
```

Following is the implementation to sort an array of size n where elements are in range from 0 to n2 – 1.

## C++

 `#include ` `using` `namespace` `std; ` ` `  `// A function to do counting sort of arr[] according to ` `// the digit represented by exp. ` `int` `countSort(``int` `arr[], ``int` `n, ``int` `exp``) ` `{ ` ` `  `    ``int` `output[n]; ``// output array ` `    ``int` `i, count[n] ; ` `    ``for` `(``int` `i=0; i < n; i++) ` `       ``count[i] = 0; ` ` `  `    ``// Store count of occurrences in count[] ` `    ``for` `(i = 0; i < n; i++) ` `        ``count[ (arr[i]/``exp``)%n ]++; ` ` `  `    ``// Change count[i] so that count[i] now contains actual ` `    ``// position of this digit in output[] ` `    ``for` `(i = 1; i < n; i++) ` `        ``count[i] += count[i - 1]; ` ` `  `    ``// Build the output array ` `    ``for` `(i = n - 1; i >= 0; i--) ` `    ``{ ` `        ``output[count[ (arr[i]/``exp``)%n] - 1] = arr[i]; ` `        ``count[(arr[i]/``exp``)%n]--; ` `    ``} ` ` `  `    ``// Copy the output array to arr[], so that arr[] now ` `    ``// contains sorted numbers according to current digit ` `    ``for` `(i = 0; i < n; i++) ` `        ``arr[i] = output[i]; ` `} ` ` `  ` `  `// The main function to that sorts arr[] of size n using Radix Sort ` `void` `sort(``int` `arr[], ``int` `n) ` `{ ` `    ``// Do counting sort for first digit in base n. Note that ` `    ``// instead of passing digit number, exp (n^0 = 1) is passed. ` `    ``countSort(arr, n, 1); ` ` `  `    ``// Do counting sort for second digit in base n. Note that ` `    ``// instead of passing digit number, exp (n^1 = n) is passed. ` `    ``countSort(arr, n, n); ` `} ` ` `  `// A utility function to print an array ` `void` `printArr(``int` `arr[], ``int` `n) ` `{ ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``cout << arr[i] << ``" "``; ` `} ` ` `  `// Driver program to test above functions ` `int` `main() ` `{ ` `    ``// Since array size is 7, elements should be from 0 to 48 ` `    ``int` `arr[] = {40, 12, 45, 32, 33, 1, 22}; ` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr[0]); ` `    ``cout << ``"Given array is n"``; ` `    ``printArr(arr, n); ` ` `  `    ``sort(arr, n); ` ` `  `    ``cout << ``"nSorted array is n"``; ` `    ``printArr(arr, n); ` `    ``return` `0; ` `}`

## Java

 `// Java program to sort an array of size n where elements are ` `// in range from 0 to n^2 – 1. ` `class` `Sort1ToN2 ` `{ ` `    ``// A function to do counting sort of arr[] according to ` `    ``// the digit represented by exp. ` `    ``void` `countSort(``int` `arr[], ``int` `n, ``int` `exp) ` `    ``{ ` `        ``int` `output[] = ``new` `int``[n]; ``// output array ` `        ``int` `i, count[] = ``new` `int``[n] ; ` `        ``for` `(i=``0``; i < n; i++) ` `           ``count[i] = ``0``; ` ` `  `        ``// Store count of occurrences in count[] ` `        ``for` `(i = ``0``; i < n; i++) ` `            ``count[ (arr[i]/exp)%n ]++; ` ` `  `        ``// Change count[i] so that count[i] now contains actual ` `        ``// position of this digit in output[] ` `        ``for` `(i = ``1``; i < n; i++) ` `            ``count[i] += count[i - ``1``]; ` ` `  `        ``// Build the output array ` `        ``for` `(i = n - ``1``; i >= ``0``; i--) ` `        ``{ ` `            ``output[count[ (arr[i]/exp)%n] - ``1``] = arr[i]; ` `            ``count[(arr[i]/exp)%n]--; ` `        ``} ` ` `  `        ``// Copy the output array to arr[], so that arr[] now ` `        ``// contains sorted numbers according to current digit ` `        ``for` `(i = ``0``; i < n; i++) ` `            ``arr[i] = output[i]; ` `    ``} ` ` `  ` `  `    ``// The main function to that sorts arr[] of size n using Radix Sort ` `    ``void` `sort(``int` `arr[], ``int` `n) ` `    ``{ ` `        ``// Do counting sort for first digit in base n. Note that ` `        ``// instead of passing digit number, exp (n^0 = 1) is passed. ` `        ``countSort(arr, n, ``1``); ` ` `  `        ``// Do counting sort for second digit in base n. Note that ` `        ``// instead of passing digit number, exp (n^1 = n) is passed. ` `        ``countSort(arr, n, n); ` `    ``} ` ` `  `    ``// A utility function to print an array ` `    ``void` `printArr(``int` `arr[], ``int` `n) ` `    ``{ ` `        ``for` `(``int` `i = ``0``; i < n; i++) ` `            ``System.out.print(arr[i]+``" "``); ` `    ``} ` ` `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``Sort1ToN2 ob = ``new` `Sort1ToN2(); ` ` `  `        ``// Since array size is 7, elements should be from 0 to 48 ` `        ``int` `arr[] = {``40``, ``12``, ``45``, ``32``, ``33``, ``1``, ``22``}; ` `        ``int` `n = arr.length; ` `        ``System.out.println(``"Given array"``); ` `        ``ob.printArr(arr, n); ` ` `  `        ``ob.sort(arr, n); ` ` `  `        ``System.out.println(``"Sorted array"``); ` `        ``ob.printArr(arr, n); ` `    ``} ` `} ` `/*This code is contributed by Rajat Mishra */`

## Python3

 `# Python3 the implementation to sort an  ` `# array of size n ` ` `  `# A function to do counting sort of arr[]  ` `# according to the digit represented by exp.  ` `def` `countSort(arr, n, exp):  ` `    ``output ``=` `[``0``] ``*` `n ``# output array  ` `    ``count ``=` `[``0``] ``*` `n ` `    ``for` `i ``in` `range``(n): ` `        ``count[i] ``=` `0` ` `  `    ``# Store count of occurrences in count[]  ` `    ``for` `i ``in` `range``(n): ` `        ``count[ (arr[i] ``/``/` `exp) ``%` `n ] ``+``=` `1` ` `  `    ``# Change count[i] so that count[i] now contains  ` `    ``# actual position of this digit in output[]  ` `    ``for` `i ``in` `range``(``1``, n):  ` `        ``count[i] ``+``=` `count[i ``-` `1``]  ` ` `  `    ``# Build the output array  ` `    ``for` `i ``in` `range``(n ``-` `1``, ``-``1``, ``-``1``):  ` ` `  `        ``output[count[ (arr[i] ``/``/` `exp) ``%` `n] ``-` `1``] ``=` `arr[i]  ` `        ``count[(arr[i] ``/``/` `exp) ``%` `n] ``-``=` `1` ` `  `    ``# Copy the output array to arr[], so that  ` `    ``# arr[] now contains sorted numbers according  ` `    ``# to current digit  ` `    ``for` `i ``in` `range``(n):  ` `        ``arr[i] ``=` `output[i]  ` ` `  `# The main function to that sorts arr[] of  ` `# size n using Radix Sort  ` `def` `sort(arr, n) : ` `     `  `    ``# Do counting sort for first digit in base n.  ` `    ``# Note that instead of passing digit number, ` `    ``# exp (n^0 = 1) is passed.  ` `    ``countSort(arr, n, ``1``)  ` ` `  `    ``# Do counting sort for second digit in base n.  ` `    ``# Note that instead of passing digit number,  ` `    ``# exp (n^1 = n) is passed.  ` `    ``countSort(arr, n, n)  ` ` `  `# Driver Code ` `if` `__name__ ``=``=``"__main__"``:  ` `     `  `    ``# Since array size is 7, elements should ` `    ``# be from 0 to 48  ` `    ``arr ``=` `[``40``, ``12``, ``45``, ``32``, ``33``, ``1``, ``22``] ` `    ``n ``=` `len``(arr)  ` `    ``print``(``"Given array is"``) ` `    ``print``(``*``arr) ` `     `  `    ``sort(arr, n) ` `     `  `    ``print``(``"Sorted array is"``) ` `    ``print``(``*``arr) ` ` `  `# This code is contribute by  ` `# Shubham Singh(SHUBHAMSINGH10) `

## C#

 `// C# program to sort an array of  ` `// size n where elements are ` `// in range from 0 to n^2 – 1. ` `using` `System; ` ` `  `class` `GFG { ` `     `  `    ``// A function to do counting  ` `    ``// sort of arr[] according to ` `    ``// the digit represented by exp. ` `    ``static` `void` `countSort(``int``[] arr,  ` `                          ``int` `n,  ` `                          ``int` `exp) ` `    ``{ ` `         `  `        ``// output array ` `        ``int``[] output = ``new` `int``[n];  ` `        ``int``[] count = ``new` `int``[n] ; ` `        ``int` `i; ` `        ``for` `(i = 0; i < n; i++) ` `        ``count[i] = 0; ` ` `  `        ``// Store count of  ` `        ``// occurrences in count[] ` `        ``for` `(i = 0; i < n; i++) ` `            ``count[(arr[i] / exp) % n ]++; ` ` `  `        ``// Change count[i] so that  ` `        ``// count[i] now contains actual ` `        ``// position of this digit in output[] ` `        ``for` `(i = 1; i < n; i++) ` `            ``count[i] += count[i - 1]; ` ` `  `        ``// Build the output array ` `        ``for` `(i = n - 1; i >= 0; i--) ` `        ``{ ` `            ``output[count[(arr[i] /  ` `                   ``exp) % n] - 1] = arr[i]; ` `            ``count[(arr[i] / exp) % n]--; ` `        ``} ` ` `  `        ``// Copy the output array to  ` `        ``// arr[], so that arr[] now ` `        ``// contains sorted numbers  ` `        ``// according to current digit ` `        ``for` `(i = 0; i < n; i++) ` `            ``arr[i] = output[i]; ` `    ``} ` ` `  ` `  `    ``// The main function to that  ` `    ``// sorts arr[] of size n  ` `    ``// using Radix Sort ` `    ``static` `void` `sort(``int``[] arr, ``int` `n) ` `    ``{ ` `         `  `        ``// Do counting sort for first  ` `        ``// digit in base n. Note that ` `        ``// instead of passing digit number,  ` `        ``// exp (n^0 = 1) is passed. ` `        ``countSort(arr, n, 1); ` ` `  `        ``// Do counting sort for second  ` `        ``// digit in base n. Note that ` `        ``// instead of passing digit number,  ` `        ``// exp (n^1 = n) is passed. ` `        ``countSort(arr, n, n); ` `    ``} ` ` `  `    ``// A utility function ` `    ``// to print an array ` `    ``static` `void` `printArr(``int``[] arr, ``int` `n) ` `    ``{ ` `        ``for` `(``int` `i = 0; i < n; i++) ` `            ``Console.Write(arr[i] + ``" "``); ` `    ``} ` ` `  `    ``// Driver Code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `         `  `        ``// Since array size is 7,  ` `        ``// elements should be  ` `        ``// from 0 to 48 ` `        ``int``[] arr = {40, 12, 45, 32, 33, 1, 22}; ` `        ``int` `n = arr.Length; ` `        ``Console.WriteLine(``"Given array"``); ` `        ``printArr(arr, n); ` ` `  `        ``sort(arr, n); ` ` `  `        ``Console.WriteLine(``"\nSorted array"``); ` `        ``printArr(arr, n); ` `    ``} ` `} ` ` `  `// This code is contributed by Ajit. `

Output:

```Given array is
40 12 45 32 33 1 22
Sorted array is
1 12 22 32 33 40 45```

How to sort if range is from 1 to n2?
If range is from 1 to n n2, the above process can not be directly applied, it must be changed. Consider n = 100 and range from 1 to 10000. Since the base is 100, a digit must be from 0 to 99 and there should be 2 digits in the numbers. But the number 10000 has more than 2 digits. So to sort numbers in a range from 1 to n2, we can use following process.
1) Subtract all numbers by 1.
2) Since the range is now 0 to n2, do counting sort twice as done in the above implementation.
3) After the elements are sorted, add 1 to all numbers to obtain the original numbers.

How to sort if range is from 0 to n^3 -1?
Since there can be 3 digits in base n, we need to call counting sort 3 times.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up

Article Tags :
Practice Tags :

6

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.