Minimum number of insertions required such that first K natural numbers can be obtained as sum of a subsequence of the array

Given an array arr[] consisting of N positive integers and a positive integer K, the task is to find the minimum number of elements that are required to be inserted such that all numbers from the range [1, K] can be obtained as the sum of any subsequence of the array.

Examples:

Input: arr[] = {1, 3, 5}, K = 10
Output: 1
Explanation:
Appending the element {1} to the array modifies the array to {1, 3, 5, 1}. Now the all the sum over the range [1, K] can be obtained as:

1. Sum 1: The elements are {1}.
2. Sum 2: The elements are {1, 1}.
3. Sum 3: The elements are {3}.
4. Sum 4: The elements are {1. 3}.
5. Sum 5: The elements are {1, 3, 1}.
6. Sum 6: The elements are {1, 5}.
7. Sum 7: The elements are {1, 5, 1}.
8. Sum 8: The elements are {3, 5}.
9. Sum 9: The elements are {1, 3, 5}.
10. Sum 10: The elements are {1, 3, 5, 1}.

Input: arr[] = {2, 6, 8, 12, 19}, K = 20
Output: 2

Approach: The given problem can be solved by sorting the array in increasing order and then try to make the sum value over the range [1, K] using the fact that if the sum of array elements X, then all the values over the range [1, X] can be formed. Otherwise, it is required to insert the value (sum + 1) as an array element. Follow the steps below to solve the problem:

• Sort the array arr[] in increasing order.
• Initialize the variable, say index as 0 to maintain the index of the array element and count as 0 to store the resultant total elements added.
• If the value of arr[0] is greater than 1, then 1 needs to be appended, so increase the value of the count by 1. Otherwise, increase the value of the index by 1.
• Initialize the variable, say expect as 2 to maintain the next value expected in the range from 1 to K to be formed from the array arr[].
• Iterate a loop until the value of expect is at most K and perform the following steps:
  • If the index is greater than equal to N or arr[index] is greater than the value of expect, then increase the value of count by 1 and multiply the value of expect by 2.
  • Otherwise, increase the value of expect by arr[index] and increase the value of the index by 1.
• After completing the above steps, print the value of count as the result.

Below is the implementation of the above approach.

C++14

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to find the minimum number of elements that must
// be added to make the sum of array element over the range
// [1, K]
void findMinimumNumberOfElements(int n, int k, int arr[])
{
    // Sort the given array
    sort(arr, arr + n);

    // Stores the index for the array
    int index = 0, count = 0;
    // If 1 is not present, then append it
    if (arr[0] > 1)
        ++count;
    // Move on to next index
    else
        ++index;

    // The expected value in the array
    long long expect = 2;
    while (expect <= k) {
        // Need to append this number
        if (index >= n || arr[index] > expect) {
            ++count;
            expect += expect;
        }

        // Otherwise, expand the range by current number
        else {
            expect += arr[index];
            ++index;
        }
    }
    // Print the answer
    cout << count;
}

// Driver Code
int main()
{
    int arr[] = { 2, 6, 8, 12, 19 };
    int K = 20;
    int N = sizeof(arr) / sizeof(arr[0]);
    findMinimumNumberOfElements(N, K, arr);

    return 0;
}

// This code is contributed by Aditya Kumar (adityakumar129)

C

// C program for the above approach

#include <stdio.h>
#include <stdlib.h>

int cmpfunc(const void* a, const void* b)
{
    return (*(int*)a - *(int*)b);
}

// Function to find the minimum number of elements that must
// be added to make the sum of array element over the range
// [1, K]
void findMinimumNumberOfElements(int n, int k, int arr[])
{
    // Sort the given array
    qsort(arr, n, sizeof(int), cmpfunc);

    // Stores the index for the array
    int index = 0, count = 0;
    // If 1 is not present, then append it
    if (arr[0] > 1)
        ++count;
    // Move on to next index
    else
        ++index;

    // The expected value in the array
    long long expect = 2;
    while (expect <= k) {
        // Need to append this number
        if (index >= n || arr[index] > expect) {
            ++count;
            expect += expect;
        }

        // Otherwise, expand the range by current number
        else {
            expect += arr[index];
            ++index;
        }
    }
    // Print the answer
    printf("%d", count);
}

// Driver Code
int main()
{
    int arr[] = { 2, 6, 8, 12, 19 };
    int K = 20;
    int N = sizeof(arr) / sizeof(arr[0]);
    findMinimumNumberOfElements(N, K, arr);

    return 0;
}

// This code is contributed by Aditya Kumar (adityakumar129)

Java

// Java program for the above approach
import java.io.*;
import java.util.Arrays;

class GFG {

    // Function to find the minimum number of elements that
    // must be added to make the sum of array element over
    // the range [1, K]
    static void findMinimumNumberOfElements(int n, int k, int[] arr)
    {
        // Sort the given array
        Arrays.sort(arr);

        // Stores the index for the array
        int index = 0, count = 0;
        // If 1 is not present, then append it

        if (arr[0] > 1)
            ++count;
        // Move on to next index
        else
            ++index;

        // The expected value in the array
        long expect = 2;
        while (expect <= k) {

            // Need to append this number
            if (index >= n || arr[index] > expect) {
                ++count;
                expect += expect;
            }

            // Otherwise, expand the range by current number
            else {
                expect += arr[index];
                ++index;
            }
        }

        // Print the answer
        System.out.println(count);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 2, 6, 8, 12, 19 };
        int K = 20;
        int N = arr.length;
        findMinimumNumberOfElements(N, K, arr);
    }
}

// This code is contributed by Aditya Kumar (adityakumar129)

Python3

# Python 3 program for the above approach

# Function to find the minimum number
# of elements that must be added to
# make the sum of array element over
# the range [1, K]
def findMinimumNumberOfElements(n, k, arr):
    # Sort the given array
    arr.sort()

    # Stores the index for the
    # array
    index = 0
    count = 0

    if (arr[0] > 1):
        # If 1 is not present, then
        # append it
        count += 1

    # Move on to next index
    else:
        index += 1

    # The expected value in the array
    expect = 2
    while (expect <= k):

        # Need to append this number
        if (index >= n or arr[index] > expect):
            count += 1
            expect += expect

        # Otherwise, expand the range
        # by current number
        else:
            expect += arr[index]
            index += 1

    # Print the answer
    print(count)

# Driver Code
if __name__ == '__main__':
    arr = [2, 6, 8, 12, 19]
    K = 20
    N = len(arr)
    findMinimumNumberOfElements(N, K, arr)

    # This code is contributed by ipg2016107.

C#

// C# program for the above approach
using System;

class GFG {

 // Function to find the minimum number
    // of elements that must be added to
    // make the sum of array element over
    // the range [1, K]
    static void findMinimumNumberOfElements(int n, int k,
                                            int[] arr)
    {
        // Sort the given array
        Array.Sort(arr);

        // Stores the index for the
        // array
        int index = 0;
        int count = 0;

        if (arr[0] > 1) {

            // If 1 is not present, then
            // append it
            ++count;
        }

        // Move on to next index
        else {
            ++index;
        }

        // The expected value in the array
        long expect = 2;
        while (expect <= k) {

            // Need to append this number
            if (index >= n || arr[index] > expect) {
                ++count;
                expect += expect;
            }

            // Otherwise, expand the range
            // by current number
            else {
                expect += arr[index];
                ++index;
            }
        }

        // Print the answer
        Console.WriteLine(count);
    }

    // Driver code
    public static void Main()
    {
        int[] arr = { 2, 6, 8, 12, 19 };
        int K = 20;
        int N = arr.Length;
        findMinimumNumberOfElements(N, K, arr);
    }
}

// This code is contributed by avijitmondal1998.

Javascript

<script>
// Javascript program for the above approach

// Function to find the minimum number
// of elements that must be added to
// make the sum of array element over
// the range [1, K]
function findMinimumNumberOfElements(n, k, arr) {
    // Sort the given array
    arr.sort((a, b) => a - b);

    // Stores the index for the
    // array
    let index = 0;
    let count = 0;

    if (arr[0] > 1) {

        // If 1 is not present, then
        // append it
        ++count;
    }

    // Move on to next index
    else {
        ++index;
    }

    // The expected value in the array
    let expect = 2;
    while (expect <= k) {

        // Need to append this number
        if (index >= n || arr[index] > expect) {
            ++count;
            expect += expect;
        }

        // Otherwise, expand the range
        // by current number
        else {
            expect += arr[index];
            ++index;
        }
    }

    // Print the answer
    document.write(count);
}

// Driver Code

let arr = [2, 6, 8, 12, 19];
let K = 20;
let N = arr.length;
findMinimumNumberOfElements(N, K, arr);

// This code is contributed by _saurabh_jaiswal.
</script>

Output:

2

Time Complexity: O(max(K, N*log N))
Auxiliary Space: O(1)