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K-th smallest element after removing some integers from natural numbers

Given an array arr[] of size ‘n’ and a positive integer k. Consider series of natural numbers and remove arr[0], arr[1], arr[2], …, arr[p] from it. Now the task is to find k-th smallest number in the remaining set of natural numbers. If no such number exists print “-1”.

Examples :

Input : arr[] = { 1 } and k = 1.
Output: 2
Natural numbers are {1, 2, 3, 4, .... }
After removing {1}, we get {2, 3, 4, ...}.
Now, K-th smallest element = 2.

Input : arr[] = {1, 3}, k = 4.
Output : 6
First 5 Natural number {1, 2, 3, 4, 5, 6,  .. }
After removing {1, 3}, we get {2, 4, 5, 6, ... }.



Method 1 (Simple):
Make an auxiliary array b[] for presence/absence of natural numbers and initialize all with 0. Make all the integer equal to 1 which are present in array arr[] i.e b[arr[i]] = 1. Now, run a loop and decrement k whenever unmarked cell is encountered. When the value of k is 0, we get the answer.

Below is implementation of this approach:

C++

// C++ program to find the K-th smallest element
// after removing some integers from natural number.
#include<bits/stdc++.h>
#define MAX 1000000
using namespace std;
  
// Return the K-th smallest element.
int ksmallest(int arr[], int n, int k)
{
    // Making an array, and mark all number as unmarked.
    int b[MAX];
    memset(b, 0, sizeof b);
  
    // Marking the number present in the given array.
    for (int i = 0; i < n; i++)
        b[arr[i]] = 1;
  
    for (int j=1; j<MAX; j++)
    {
        // If j is unmarked, reduce k by 1.
        if (b[j] != 1)
            k--;
  
        // If k is 0 return j.
        if (!k)
            return j;
    }
}
  
// Driven Program
int main()
{
    int k = 1;
    int arr[] = { 1 };
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << ksmallest(arr, n, k);
    return 0;
}

/div>

Java

// Java program to find the K-th smallest element 
// after removing some integers from natural number.
class GFG
{
  
    static final int MAX = 1000000;
  
    // Return the K-th smallest element. 
    static int ksmallest(int arr[], int n, int k)
    {
        // Making an array, and mark 
        // all number as unmarked. 
        int b[] = new int[MAX];
  
        // Marking the number present
        // in the given array. 
        for (int i = 0; i < n; i++)
        {
            b[arr[i]] = 1;
        }
  
        for (int j = 1; j < MAX; j++)
        {
            // If j is unmarked, reduce k by 1. 
            if (b[j] != 1)
            {
                k--;
            }
  
            // If k is 0 return j. 
            if (k != 1)
            {
                return j;
            }
        }
        return Integer.MAX_VALUE;
    }
  
    // Driven code 
    public static void main(String[] args) 
    {
        int k = 1;
        int arr[] = {1};
        int n = arr.length;
        System.out.println(ksmallest(arr, n, k));
    }
}
  
// This code has been contributed by 29AjayKumar

Python3

# Python program to find the K-th smallest element
# after removing some integers from natural number.
MAX = 1000000
  
  
# Return the K-th smallest element.
def ksmallest(arr, n, k):
      
    # Making an array, and mark all number as unmarked.
    b = [0]*MAX;
  
    # Marking the number present in the given array.
    for i in range(n):
        b[arr[i]] = 1;
  
    for j in range(1,MAX):
        # If j is unmarked, reduce k by 1.
        if (b[j] != 1):
            k-=1;
  
        # If k is 0 return j.
        if (k is not 1):
            return j;
              
# Driven Program
k = 1;
arr = [ 1 ];
n = len(arr);
print(ksmallest(arr, n, k));
  
# This code contributed by Rajput-Ji

C#

// C# program to find the K-th smallest element 
// after removing some integers from natural number.
using System;
  
class GFG
{
  
    static int MAX = 1000000;
  
    // Return the K-th smallest element. 
    static int ksmallest(int []arr, int n, int k)
    {
        // Making an array, and mark 
        // all number as unmarked. 
        int []b = new int[MAX];
  
        // Marking the number present
        // in the given array. 
        for (int i = 0; i < n; i++)
        {
            b[arr[i]] = 1;
        }
  
        for (int j = 1; j < MAX; j++)
        {
            // If j is unmarked, reduce k by 1. 
            if (b[j] != 1)
            {
                k--;
            }
  
            // If k is 0 return j. 
            if (k != 1)
            {
                return j;
            }
        }
        return int.MaxValue;
    }
  
    // Driven code 
    public static void Main() 
    {
        int k = 1;
        int []arr = {1};
        int n = arr.Length;
        Console.WriteLine(ksmallest(arr, n, k));
    }
}
  
/* This code contributed by PrinciRaj1992 */


Output :

2

Time Complexity : O(n).

 

Method 2 (Efficient):
First, sort the array arr[]. Observe, there will be arr[0] – 1 numbers between 0 and arr[0], similarly, arr[1] – arr[0] – 1 numbers between arr[0] and arr[1] and so on. So, if k lies between arr[i] – arr[i+1] – 1, then return K-th smallest element in the range. Else reduce k by arr[i] – arr[i+1] – 1 i.e., k = k – (arr[i] – arr[i+1] – 1).

Algorithm to solve the problem:

1. Sort the array arr[].
2. For i = 1 to k. Find c = arr[i+1] - arr[i] -1.
  a) if k - c <= 0, return arr[i-1] + k.
  b) else k = k - c.
Below is implementation of this approach:

C++

// C++ program to find the Kth smallest element
// after removing some integer from first n
// natural number.
#include<bits/stdc++.h>
using namespace std;
  
// Return the K-th smallest element.
int ksmallest(int arr[], int n, int k)
{
    sort(arr, arr+n);
  
    // Checking if k lies before 1st element
    if (k < arr[0])
        return k;
  
    // If k is the first element of array arr[].
    if (k == arr[0])
        return arr[0] + 1;
  
    // If k is more than last element
    if (k > arr[n-1])
        return k + n;
  
    // If first element of array is 1.
    if (arr[0] == 1)
        k--;
  
    // Reducing k by numbers before arr[0].
    else
        k -= (arr[0] - 1);
  
    // Finding k'th smallest element after removing
    // array elements.
    for (int i=1; i<n; i++)
    {
        // Finding count of element between i-th
        // and (i-1)-th element.
        int c = arr[i] - arr[i-1] - 1;
        if (k <= c)
            return arr[i-1] + k;
        else
            k -= c;
    }
  
    return arr[n-1] + k;
}
  
// Driven Program
int main()
{
    int k = 1;
    int arr[] = { 1 };
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << ksmallest(arr, n, k);
    return 0;
}

Java

// Java program to find the 
// Kth smallest element after 
// removing some integer from 
// first n natural number.
import java.util.Arrays;
import java.io.*;
  
class GFG 
{
      
// Return the K-th
// smallest element.
static int ksmallest(int arr[],
                     int n, int k)
{
    // sort(arr, arr+n);
    Arrays.sort(arr);
  
    // Checking if k lies 
    // before 1st element
    if (k < arr[0])
        return k;
  
    // If k is the first 
    // element of array arr[].
    if (k == arr[0])
        return arr[0] + 1;
  
    // If k is more
    // than last element
    if (k > arr[n - 1])
        return k + n;
  
    // If first element 
    // of array is 1.
    if (arr[0] == 1)
        k--;
  
    // Reducing k by numbers
    // before arr[0].
    else
        k -= (arr[0] - 1);
  
    // Finding k'th smallest 
    // element after removing
    // array elements.
    for (int i = 1; i < n; i++)
    {
        // Finding count of
        // element between i-th
        // and (i-1)-th element.
        int c = arr[i] - arr[i - 1] - 1;
        if (k <= c)
            return arr[i - 1] + k;
        else
            k -= c;
    }
  
    return arr[n-1] + k;
}
  
// Driven Code
public static void main (String[] args) 
{
    int k = 1;
    int arr[] = { 1 };
    int n = arr.length;
    System.out.println(ksmallest(arr, n, k));
}
}
  
// This code is contributed 
// by ajit

C#

// C# program to find the 
// Kth smallest element after 
// removing some integer from 
// first n natural number.
using System;
  
class GFG
{
// Return the K-th
// smallest element.
static int ksmallest(int []arr,
                     int n, int k)
{
    // sort(arr, arr+n);
    Array.Sort(arr);
  
    // Checking if k lies 
    // before 1st element
    if (k < arr[0])
        return k;
  
    // If k is the first 
    // element of array arr[].
    if (k == arr[0])
        return arr[0] + 1;
  
    // If k is more
    // than last element
    if (k > arr[n - 1])
        return k + n;
  
    // If first element 
    // of array is 1.
    if (arr[0] == 1)
        k--;
  
    // Reducing k by numbers
    // before arr[0].
    else
        k -= (arr[0] - 1);
  
    // Finding k'th smallest 
    // element after removing
    // array elements.
    for (int i = 1; i < n; i++)
    {
        // Finding count of
        // element between i-th
        // and (i-1)-th element.
        int c = arr[i] - 
                arr[i - 1] - 1;
        if (k <= c)
            return arr[i - 1] + k;
        else
            k -= c;
    }
  
    return arr[n-1] + k;
}
  
// Driver Code
static public void Main ()
{
    int k = 1;
    int []arr = { 1 };
    int n = arr.Length;
    Console.WriteLine(ksmallest(arr, n, k));
}
}
  
// This code is contributed 
// by ajit

PHP

<?php
// PHP program to find the Kth 
// smallest element after 
// removing some integer from 
// first n natural number.
  
// Return the K-th 
// smallest element.
function ksmallest($arr, $n, $k)
{
    sort($arr);
  
    // Checking if k lies
    // before 1st element
    if ($k < $arr[0])
        return $k;
  
    // If k is the first 
    // element of array arr[].
    if ($k == $arr[0])
        return $arr[0] + 1;
  
    // If k is more 
    // than last element
    if ($k > $arr[$n - 1])
        return $k + $n;
  
    // If first element 
    // of array is 1.
    if ($arr[0] == 1)
        $k--;
  
    // Reducing k by numbers
    // before arr[0].
    else
        $k -= ($arr[0] - 1);
  
    // Finding k'th smallest element 
    // after removing array elements.
    for ($i = 1; $i < $n; $i++)
    {
        // Finding count of element between 
        // i-th and (i-1)-th element.
        $c = $arr[$i] - $arr[$i - 1] - 1;
        if ($k <= $c)
            return $arr[$i - 1] + $k;
        else
            $k -= $c;
    }
  
    return $arr[$n - 1] + $k;
}
  
// Driver Code
$k = 1;
$arr = array ( 1 );
$n = sizeof($arr);
echo ksmallest($arr, $n, $k);
  
// This code is contributed by aj_36
?>


Output :

2

More efficient method : K-th smallest element after removing given integers from natural numbers | Set 2

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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