Arrange given numbers to form the smallest number

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

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>

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

?>

Output

1212569

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;
}

Java

import java.util.Arrays;

public class SmallestNumber {

    public static boolean compare(String a, String b) {
        return (a + b).compareTo(b + a) < 0;
    }

    public static int partition(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++;
                String temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
            }
        }
        String temp = arr[i + 1];
        arr[i + 1] = arr[high];
        arr[high] = temp;
        return i + 1;
    }

    public static void quicksort(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);
        }
    }

    public static String smallestNumber(int[] nums) {
        String[] numStrs = new String[nums.length];
        for (int i = 0; i < nums.length; i++) {
            numStrs[i] = String.valueOf(nums[i]);
        }
        quicksort(numStrs, 0, numStrs.length - 1);
        StringBuilder result = new StringBuilder();
        for (String numStr : numStrs) {
            result.append(numStr);
        }
        return result.toString();
    }

    public static void main(String[] args) {
        int[] nums = {5, 6, 2, 9, 21, 1};
        System.out.println(smallestNumber(nums)); // Output: 3033459
    }
}

Python3

def compare(a, b):
    return (a + b) < (b + a)

def partition(arr, low, high):
    pivot = arr[high]
    i = low - 1
    for j in range(low, high):
        if compare(arr[j], pivot):
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

def quicksort(arr, low, high):
    if low < high:
        pi = partition(arr, low, high)
        quicksort(arr, low, pi - 1)
        quicksort(arr, pi + 1, high)

def smallestNumber(nums):
    numStrs = [str(num) for num in nums]
    quicksort(numStrs, 0, len(numStrs) - 1)
    result = "".join(numStrs)
    return result

if __name__ == "__main__":
    nums = [5, 6, 2, 9, 21, 1]
    print(smallestNumber(nums))  # Output: 3033459

C#

using System;
using System.Collections.Generic;

class Program {
    // Function to compare two strings based on their
    // concatenated values
    static bool Compare(string a, string b)
    {
        return (a + b).CompareTo(b + a) < 0;
    }

    // Function to partition the array for quicksort
    static int Partition(List<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, j);
            }
        }

        Swap(arr, i + 1, high);
        return i + 1;
    }

    // Function to perform quicksort on the array
    static void QuickSort(List<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);
        }
    }

    // Function to find the smallest number by concatenating
    // array elements
    static string SmallestNumber(List<int> nums)
    {
        List<string> numStrs = new List<string>();
        foreach(int num in nums)
        {
            numStrs.Add(num.ToString());
        }

        QuickSort(numStrs, 0, numStrs.Count - 1);

        string result = "";
        foreach(string numStr in numStrs)
        {
            result += numStr;
        }

        return result;
    }

    // Function to swap two elements in a list
    static void Swap(List<string> arr, int i, int j)
    {
        string temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }

    // Main function to test the SmallestNumber function
    static void Main()
    {
        List<int> nums = new List<int>{ 5, 6, 2, 9, 21, 1 };
        Console.WriteLine(
            SmallestNumber(nums)); // Output: 3033459
    }
}

Javascript

// Function to compare two strings for sorting
function compare(a, b) {
    return (a + b) < (b + a);
}

// Function to perform partitioning for quicksort
function partition(arr, low, high) {
    const pivot = arr[high];
    let i = low - 1;

    // Iterate through the array
    for (let j = low; j <= high - 1; j++) {
        // Check if arr[j] is less than pivot
        if (compare(arr[j], pivot)) {
            i++;

            // Swap arr[i] and arr[j]
            [arr[i], arr[j]] = [arr[j], arr[i]];
        }
    }

    // Swap arr[i+1] and arr[high]
    [arr[i + 1], arr[high]] = [arr[high], arr[i + 1]];
    return i + 1;
}

// Function to implement quicksort algorithm
function quicksort(arr, low, high) {
    if (low < high) {
        const pi = partition(arr, low, high);
        // Recursively sort elements before and after partition
        quicksort(arr, low, pi - 1);
        quicksort(arr, pi + 1, high);
    }
}

// Function to find the smallest number by sorting array of integers
function smallestNumber(nums) {
    // Convert integers to strings for sorting
    const numStrs = nums.map(num => String(num));
    // Sort the strings using custom comparison function
    quicksort(numStrs, 0, numStrs.length - 1);
    // Concatenate sorted strings to form the result
    const result = numStrs.join('');
    return result;
}

// Test case
const nums = [5, 6, 2, 9, 21, 1];
console.log(smallestNumber(nums)); // Output: 3033459

Output

1212569

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