Arrange given numbers to form the smallest number

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Given an array arr[] of integer elements, the task is to arrange them in such a way that these numbers form the smallest possible number.
For example, if the given array is {5, 6, 2, 9, 21, 1} then the arrangement will be 1212569.

Examples:

Input: arr[] = {5, 6, 2, 9, 21, 1}
Output: 1212569

Input: arr[] = {1, 2, 1, 12, 33, 211, 50}
Output: 111221123350

Approach: If all the given numbers are of at most one digit then the simple approach is sorting all numbers in ascending order. But if there is some number which have more than a single-digit then this approach will not work.
Therefore, we have to sort the array by comparing any two elements in the following way:
If the elements are A and B, then compare (A + B) with (B + A) where + represents concatenation.
Below is the implementation of the above approach:

C++

// C++ implementation of the approach
#include <algorithm>
#include <iostream>
using namespace std;
 
// Utility function to print
// the contents of an array
void printArr(int arr[], int n)
{
    for (int i = 0; i < n; i++)
        cout << arr[i];
}
 
// A comparison function that return true
// if 'AB' is smaller than 'BA' when
// we concatenate two numbers 'A' and 'B'
// For example, it will return true if
// we pass 12 and 24 as arguments.
// This function will be used by sort() function
bool compare(int num1, int num2)
{
    // to_string function is predefined function
    // to convert a number in string
 
    // Convert first number to string format
    string A = to_string(num1);
 
    // Convert second number to string format
    string B = to_string(num2);
 
    // Check if 'AB' is smaller or 'BA'
    // and return bool value since
    // comparison operator '<=' returns
    // true or false
    return (A + B) <= (B + A);
}
 
// Function to print the arrangement
// with the smallest value
void printSmallest(int N, int arr[])
{
    // If we pass the name of the comparison
    // function it will sort the array
    // according to the compare function
    sort(arr, arr + N, compare);
 
    // Print the sorted array
    printArr(arr, N);
}
 
// Driver code
int main()
{
    int arr[] = { 5, 6, 2, 9, 21, 1 };
    int N = sizeof(arr) / sizeof(arr[0]);
    printSmallest(N, arr);
 
    return 0;
}

Java

// Java implementation of the approach 
class GFG
{
 
    // Utility function to print
    // the contents of an array
    public static void printArr(int[] arr, int n) 
    {
        for (int i = 0; i < n; i++)
            System.out.print(arr[i]);
    }
 
    // A comparison function that return negative
    // if 'AB' is smaller than 'BA' when
    // we concatenate two numbers 'A' and 'B'
    // For example, it will return negative value if
    // we pass 12 and 24 as arguments.
    // This function will be used during sort
    public static int compare(int num1, int num2)
    {
 
        // toString function is predefined function
        // to convert a number in string
 
        // Convert first number to string format
        String A = Integer.toString(num1);
 
        // Convert second number to string format
        String B = Integer.toString(num2);
         
        // Check if 'AB' is smaller or 'BA'
        // and return integer value
        return (A+B).compareTo(B+A);
    }
 
    // Function to print the arrangement
    // with the smallest value
    public static void printSmallest(int N, int[] arr) 
    {
 
        // Sort using compare function which
        // is defined above
        for (int i = 0; i < N; i++)
        {
            for (int j = i + 1; j < N; j++)
            {
                if (compare(arr[i], arr[j]) > 0)
                {
                    int temp = arr[i];
                    arr[i] = arr[j];
                    arr[j] = temp;
                }
            }
        }
 
        // Print the sorted array
        printArr(arr, N);
    }
 
    // Driver code
    public static void main(String[] args) 
    {
        int[] arr = { 5, 6, 2, 9, 21, 1 };
        int N = arr.length;
        printSmallest(N, arr);
    }
}
 
// This code is contributed by
// sanjeev2552

Python3

# Python3 implementation of the approach
 
# Utility function to print
# the contents of an array
def printArr(arr, n):
 
    for i in range(0, n):
        print(arr[i], end = "")
 
# A comparison function that return true
# if 'AB' is smaller than 'BA' when
# we concatenate two numbers 'A' and 'B'
# For example, it will return true if
# we pass 12 and 24 as arguments.
# This function will be used by sort() function
def compare(num1, num2):
 
    # Convert first number to string format
    A = str(num1)
 
    # Convert second number to string format
    B = str(num2)
 
    # Check if 'AB' is smaller or 'BA'
    # and return bool value since
    # comparison operator '<=' returns
    # true or false
    return int(A + B) <= int(B + A)
     
def sort(arr):
     
    for i in range(len(arr)):
        for j in range(i + 1, len(arr)):
             
            if compare(arr[i], arr[j]) == False:
                arr[i], arr[j] = arr[j], arr[i]
 
# Function to print the arrangement
# with the smallest value
def printSmallest(N, arr):
 
    # If we pass the name of the comparison
    # function it will sort the array
    # according to the compare function
    sort(arr)
 
    # Print the sorted array
    printArr(arr, N)
 
# Driver code
if __name__ == "__main__":
 
    arr = [5, 6, 2, 9, 21, 1]
    N = len(arr)
    printSmallest(N, arr)
 
# This code is contributed by Rituraj Jain

C#

// C# implementation for above approach
using System;
 
class GFG
{
 
    // Utility function to print
    // the contents of an array
    public static void printArr(int[] arr, int n) 
    {
        for (int i = 0; i < n; i++)
            Console.Write(arr[i]);
    }
 
    // A comparison function that return negative
    // if 'AB' is smaller than 'BA' when
    // we concatenate two numbers 'A' and 'B'
    // For example, it will return negative value if
    // we pass 12 and 24 as arguments.
    // This function will be used during sort
    public static int compare(int num1, int num2)
    {
 
        // toString function is predefined function
        // to convert a number in string
 
        // Convert first number to string format
        String A = num1.ToString();
 
        // Convert second number to string format
        String B = num2.ToString();
         
        // Check if 'AB' is smaller or 'BA'
        // and return integer value
        return (A+B).CompareTo(B+A);
    }
 
    // Function to print the arrangement
    // with the smallest value
    public static void printSmallest(int N, int[] arr) 
    {
 
        // Sort using compare function which
        // is defined above
        for (int i = 0; i < N; i++)
        {
            for (int j = i + 1; j < N; j++)
            {
                if (compare(arr[i], arr[j]) > 0)
                {
                    int temp = arr[i];
                    arr[i] = arr[j];
                    arr[j] = temp;
                }
            }
        }
 
        // Print the sorted array
        printArr(arr, N);
    }
 
    // Driver code
    public static void Main(String[] args) 
    {
        int[] arr = { 5, 6, 2, 9, 21, 1 };
        int N = arr.Length;
        printSmallest(N, arr);
    }
}
 
// This code is contributed by Rajput-Ji

PHP

<?php
// PHP implementation of the approach 
 
// Utility function to print 
// the contents of an array 
function printArr($arr, $n) 
{ 
    for ($i = 0; $i < $n; $i++) 
        echo $arr[$i]; 
} 
 
// A comparison function that return true 
// if 'AB' is smaller than 'BA' when 
// we concatenate two numbers 'A' and 'B' 
// For example, it will return true if 
// we pass 12 and 24 as arguments. 
// This function will be used by sort() function 
function compare($num1, $num2) 
{ 
    // to_string function is predefined function 
    // to convert a number in string 
 
    // Convert first number to string format 
    $A = (string)$num1 ; 
 
    // Convert second number to string format 
    $B = (string)$num2 ; 
 
    // Check if 'AB' is smaller or 'BA' 
    // and return bool value since 
    // comparison operator '<=' returns 
    // true or false 
    if((int)($A . $B) <= (int)($B . $A))
    {
        return true;
    }
    else
        return false;
} 
 
 
function sort_arr($arr)
{
     
    for ($i = 0; $i < count($arr) ; $i++)
    {
        for ($j = $i + 1;$j < count($arr) ; $j++)
        {
            if (compare($arr[$i], $arr[$j]) == false)
            {
                $temp = $arr[$i] ;
                $arr[$i] = $arr[$j] ;
                $arr[$j] = $temp ;
            }
        }
    }
        return $arr ;
    }
 
// Function to print the arrangement 
// with the smallest value 
function printSmallest($N, $arr) 
{ 
    // If we pass the name of the comparison 
    // function it will sort the array 
    // according to the compare function 
    $arr = sort_arr($arr); 
 
    // Print the sorted array 
    printArr($arr, $N); 
} 
 
    // Driver code 
    $arr = array(5, 6, 2, 9, 21, 1 ); 
    $N = count($arr); 
    printSmallest($N, $arr); 
 
    // This code is contributed by Ryuga
 
?>

Javascript

<script>
    // Javascript implementation of the approach
 
    // Utility function to print
    // the contents of an array
    function printArr(arr,n)
    {
        for (let i = 0; i < n; i++)
            document.write(arr[i]);
    }
 
    // A comparison function that return true
    // if 'AB' is smaller than 'BA' when
    // we concatenate two numbers 'A' and 'B'
    // For example, it will return true if
    // we pass 12 and 24 as arguments.
    // This function will be used by sort() function
    function compare(num1,num2)
    {
        // to_string function is predefined function
        // to convert a number in string
 
        // Convert first number to string format
        let A = num1.toString();
 
        // Convert second number to string format
        let B = num2.toString();
 
        // Check if 'AB' is smaller or 'BA'
        // and return bool value since
        // comparison operator '<=' returns
        // true or false
        return (A + B).localeCompare(B + A);
    }
 
    // Function to print the arrangement
    // with the smallest value
    function printSmallest(N,arr)
    {
        // If we pass the name of the comparison
        // function it will sort the array
        // according to the compare function
        // Sort using compare function which
        // is defined above
        for (let i = 0; i < N; i++)
        {
            for (let j = i + 1; j < N; j++)
            {
                if (compare(arr[i], arr[j]) > 0)
                {
                    let temp = arr[i];
                    arr[i] = arr[j];
                    arr[j] = temp;
                }
            }
        }
 
        // Print the sorted array
        printArr(arr,N);
    }
 
    // Driver code
 
    let arr = [ 5, 6, 2, 9, 21, 1 ];
    let N = arr.length;
    printSmallest(N,arr);
</script>

Output:

Time Complexity: O(nlogn)
Auxiliary Space: O(1)

Approach 2: Implementing a custom quicksort algorithm

Another way to solve the problem is to implement a custom quicksort algorithm that uses the same comparison function as in approach 1. The quicksort algorithm recursively partitions the array into two subarrays based on the comparison function, and then combines the sorted subarrays to form the final result.

C++

#include <iostream>
#include <vector>
#include <string>
 
using namespace std;
 
bool compare(string a, string b) {
    return (a+b) < (b+a);
}
 
int partition(vector<string>& arr, int low, int high) {
    string pivot = arr[high];
    int i = low - 1;
    for (int j = low; j <= high - 1; j++) {
        if (compare(arr[j], pivot)) {
            i++;
            swap(arr[i], arr[j]);
        }
    }
    swap(arr[i+1], arr[high]);
    return i+1;
}
 
void quicksort(vector<string>& arr, int low, int high) {
    if (low < high) {
        int pi = partition(arr, low, high);
        quicksort(arr, low, pi - 1);
        quicksort(arr, pi + 1, high);
    }
}
 
string smallestNumber(vector<int>& nums) {
    vector<string> numStrs;
    for (int num : nums) {
        numStrs.push_back(to_string(num));
    }
    quicksort(numStrs, 0, numStrs.size()-1);
    string result = "";
    for (string numStr : numStrs) {
        result += numStr;
    }
    return result;
}
 
int main() {
    vector<int> nums = { 5, 6, 2, 9, 21, 1 };
    cout << smallestNumber(nums) << endl; // Output: 3033459
    return 0;
}

Output

Time Complexity: O(nlogn)
Auxiliary Space: O(n)

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