Largest subset having with sum less than equal to sum of respective indices

Given an array arr[], the task is to find the length of the largest subset with the sum of elements less than or equal to the sum of its indexes(1-based indexing).

Examples:

Input: arr[] = {1, 7, 3, 5, 9, 6, 6} 
Output: 5 
Explanation: 
Largest Subset is {1, 3, 5, 6, 6} 
Sum of indexes = 1 + 3 + 4 + 6 + 7 = 21 
Sum of elements = 1 + 3 + 5 + 6 + 6 = 21

Input: arr[] = {4, 1, 6, 7, 8, 2} 
Output: 3

Naive Approach: 
The simplest approach to solve the problem is to generate all possible subsets and calculate the length of the subsets which have the sum of elements less than or equal to the sum of its respective indices. 

Time Complexity: O(N*2^N) 
Auxiliary Space: O(N)

Efficient Approach: 
Follow the steps below to solve the problem: 

• Iterate over all indices and consider only those indices whose values are greater than or equal to the values of the respective values stored in them.
• Keep updating the sum of the differences obtained in the above step.
• For the remaining elements, store their differences with their respective indexes. Sort the differences.
• Include elements into the subset one by one and subtract the difference from the sum. Keep including until an element is encountered whose difference with its index exceeds the remaining sum or all array elements have already been included.

Below is the implementation of the above approach:

// C++ Program to implement
// the above approach

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

// Function to find the length
// of the longest subset
int findSubset(int* a, int n)
{
    // Stores the sum of differences
    // between elements and
    // their respective index
    int sum = 0;

    // Stores the size of
    // the subset
    int cnt = 0;

    vector<int> v;

    // Iterate over the array
    for (int i = 1; i <= n; i++) {

        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0) {

            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }

        // Store the difference with
        // index of the remaining
        // elements
        else {
            v.push_back(a[i - 1] - i);
        }
    }

    // Sort the differences
    // in increasing order
    sort(v.begin(), v.end());

    int ptr = 0;

    // Include the differences while
    // sum remains positive or
    while (ptr < v.size()
           && sum + v[ptr] <= 0) {
        cnt += 1;
        ptr += 1;
        sum += v[ptr];
    }

    // Return the size
    return cnt;
}

// Driver Code
int main()
{
    int arr[] = { 4, 1, 6, 7,
                  8, 2 };

    int n = sizeof(arr)
            / sizeof(arr[0]);

    // Function Calling
    cout << findSubset(arr, n)
         << endl;
}

// Java program to implement 
// the above approach
import java.util.*; 
class GFG{
    
// Function to find the length 
// of the longest subset 
public static int findSubset(int[] a, int n) 
{ 
    
    // Stores the sum of differences 
    // between elements and 
    // their respective index 
    int sum = 0; 

    // Stores the size of 
    // the subset 
    int cnt = 0; 

    Vector<Integer> v = new Vector<>(); 

    // Iterate over the array 
    for(int i = 1; i <= n; i++)
    { 
        
        // If an element which is 
        // smaller than or equal 
        // to its index is encountered 
        if (a[i - 1] - i <= 0)
        { 
            
            // Increase count and sum 
            sum += a[i - 1] - i; 
            cnt += 1; 
        } 

        // Store the difference with 
        // index of the remaining 
        // elements 
        else
        { 
            v.add(a[i - 1] - i); 
        } 
    } 

    // Sort the differences 
    // in increasing order 
    Collections.sort(v); 

    int ptr = 0; 

    // Include the differences while 
    // sum remains positive or 
    while (ptr < v.size() && 
           sum + v.get(ptr) <= 0)
    { 
        cnt += 1; 
        ptr += 1; 
        sum += v.get(ptr); 
    } 

    // Return the size 
    return cnt; 
} 

// Driver code
public static void main(String[] args)
{
    int arr[] = { 4, 1, 6, 7, 8, 2 }; 
    int n = arr.length; 

    // Function Calling 
    System.out.println(findSubset(arr, n)); 
}
}

// This code is contributed by divyeshrabadiya07

# Python3 program to implement  
# the above approach 

# Function to find the length  
# of the longest subset 
def findSubset(a, n):
    
    # Stores the sum of differences  
    # between elements and  
    # their respective index  
    sum = 0 
  
    # Stores the size of  
    # the subset  
    cnt = 0
  
    v = []  
  
    # Iterate over the array  
    for i in range(1, n + 1):
        
        # If an element which is  
        # smaller than or equal  
        # to its index is encountered  
        if (a[i - 1] - i <= 0):
            
            # Increase count and sum  
            sum += a[i - 1] - i  
            cnt += 1
    
        # Store the difference with  
        # index of the remaining  
        # elements  
        else:
            v.append(a[i - 1] - i)
            
    # Sort the differences  
    # in increasing order  
    v.sort()
  
    ptr = 0
  
    # Include the differences while  
    # sum remains positive or  
    while (ptr < len(v) and 
           sum + v[ptr] <= 0):
        cnt += 1
        ptr += 1  
        sum += v[ptr]  
    
    # Return the size  
    return cnt

# Driver code
if __name__=="__main__":
    
    arr = [ 4, 1, 6, 7, 8, 2 ]  
    n = len(arr)
  
    # Function calling  
    print(findSubset(arr, n))

# This code is contributed by rutvik_56

// C# program to implement 
// the above approach
using System;
using System.Collections.Generic;

class GFG{
    
// Function to find the length 
// of the longest subset 
public static int findSubset(int[] a, int n) 
{ 
    
    // Stores the sum of differences 
    // between elements and 
    // their respective index 
    int sum = 0; 

    // Stores the size of 
    // the subset 
    int cnt = 0; 

    List<int> v = new List<int>(); 

    // Iterate over the array 
    for(int i = 1; i <= n; i++)
    { 
        
        // If an element which is 
        // smaller than or equal 
        // to its index is encountered 
        if (a[i - 1] - i <= 0)
        { 
            
            // Increase count and sum 
            sum += a[i - 1] - i; 
            cnt += 1; 
        } 

        // Store the difference with 
        // index of the remaining 
        // elements 
        else
        { 
            v.Add(a[i - 1] - i); 
        } 
    } 

    // Sort the differences 
    // in increasing order 
    v.Sort(); 

    int ptr = 0; 

    // Include the differences while 
    // sum remains positive or 
    while (ptr < v.Count && 
           sum + v[ptr] <= 0)
    { 
        cnt += 1; 
        ptr += 1; 
        sum += v[ptr]; 
    } 

    // Return the size 
    return cnt; 
} 

// Driver code
public static void Main(String[] args)
{
    int []arr = { 4, 1, 6, 7, 8, 2 }; 
    int n = arr.Length; 

    // Function calling 
    Console.WriteLine(findSubset(arr, n)); 
}
}

// This code is contributed by amal kumar choubey

<script>

// Javascript program for the above approach

// Function to find the length
// of the longest subset
function findSubset(a, n)
{
     
    // Stores the sum of differences
    // between elements and
    // their respective index
    let sum = 0;
 
    // Stores the size of
    // the subset
    let cnt = 0;
 
    let v = [];
 
    // Iterate over the array
    for(let i = 1; i <= n; i++)
    {
         
        // If an element which is
        // smaller than or equal
        // to its index is encountered
        if (a[i - 1] - i <= 0)
        {
             
            // Increase count and sum
            sum += a[i - 1] - i;
            cnt += 1;
        }
 
        // Store the difference with
        // index of the remaining
        // elements
        else
        {
            v.push(a[i - 1] - i);
        }
    }
 
    // Sort the differences
    // in increasing order
    v.sort();
 
    let ptr = 0;
 
    // Include the differences while
    // sum remains positive or
    while (ptr < v.length &&
           sum + v[ptr] <= 0)
    {
        cnt += 1;
        ptr += 1;
        sum += v[ptr];
    }
 
    // Return the size
    return cnt;
}

// Driver Code

    let arr = [ 4, 1, 6, 7, 8, 2 ];
    let n = arr.length;
 
    // Function Calling
    document.write(findSubset(arr, n));

</script>

3

Time Complexity: O(N*log(N)), the inbuilt sort function takes N log N time to complete all operations, hence the overall time taken by the algorithm is N log N
Auxiliary Space: O(N), an extra vector is used and in the worst case all elements will be stored inside it the hence algorithm takes up linear space