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Maximum subsequence sum from a given array which is a perfect square

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  • Last Updated : 11 May, 2021
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Given an array arr[], the task is to find the sum of a subsequence that forms a perfect square. If there are multiple subsequences having a sum equal to a perfect square, print the maximum sum.
Explanation: 
 

Input: arr[] = {1, 2, 3, 4, 5, 6, 7, 8, 9} 
Output: 36 
Explanation: 
Maximum possible sum which is a perfect square that can be obtained from the array is 36 obtained from the subsequence {1, 5, 6, 7, 8, 9}.
Input: arr[] = {9, 10} 
Output:
 

Naive Approach: Generate all the possible subsequences of a given array and for each subsequence, check if its sum is a Perfect Square. If such a sum is found, update the maximum sum. Finally, print the maximum sum obtained. 
Time Complexity: O(N * 2N
Auxiliary Space: O(N)
 

Efficient Approach: 
The above approach can be optimized by using Dynamic Programming approach. 
Follow the steps given below: 
 

  • Calculate the sum of the array.
  • Iterate k in the range [√sum, 0] and check if any subsequence exists with that sum k2. For every k, follow the steps below: 
    • Initialize boolean matrix subset[][] of dimensions N * sum, where sum denotes the current value of k2.
    • subset[i][j] denotes whether a subsequence of size i with sum j exists or not.
    • Initialize first column and first row as true and false respectively of subset[][].
    • Run two nested loops, outer from i in the range [1, N] and inner j in the range [1, sum]to fill the subset[][] matrix in bottom up manner based on the following conditions: 
      • if (j < arr[i – 1]), then subset[i][j] = subset[i – 1][j]
      • if (j >= arr[i – 1]), then subset[i][j] = subset[i – 1][j] || subset[i – 1][j – arr[i – 1]]
    • Finally, return the subset[n][sum].
  • The first k for which subset[n][k] is true, gives the required maximum possible subsequence sum k2.

Below is the implementation of the above approach:
 

C++




// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;
bool isSubsetSum(int arr[], int n, int sum)
{
    bool subset[n + 1][sum + 1];
 
    // If sum is 0, then answer is true
    for (int i = 0; i <= n; i++)
        subset[i][0] = true;
 
    // If sum is not 0 and arr[] is empty,
    // then answer is false
    for (int i = 1; i <= sum; i++)
        subset[0][i] = false;
 
    // Fill the subset table in bottom up manner
    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= sum; j++) {
 
            if (j < arr[i - 1])
                subset[i][j] = subset[i - 1][j];
 
            if (j >= arr[i - 1])
                subset[i][j]
                    = subset[i - 1][j]
                      || subset[i - 1][j - arr[i - 1]];
        }
    }
 
    return subset[n][sum];
}
// Function to find the sum
int findSum(int* arr, int n)
{
    int sum = 0;
    // Find sum of all values
    for (int i = 0; i < n; i++) {
        sum += arr[i];
    }
 
    int val = sqrt(sum);
 
    for (int i = val; i >= 0; i--) {
        if (isSubsetSum(arr, n, i * i)) {
            // return the value;
            return i * i;
        }
    }
 
    return 0;
}
 
// Driver Code
int main()
{
    int arr[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << findSum(arr, n) << endl;
    return 0;
}

Java




// Java program to implement
// the above approach
import java.util.*;
 
class GFG{
 
static boolean isSubsetSum(int arr[],
                           int n, int sum)
{
    boolean[][] subset = new boolean[n + 1][sum + 1];
 
    // If sum is 0, then answer is true
    for(int i = 0; i <= n; i++)
        subset[i][0] = true;
 
    // If sum is not 0 and arr[] is empty,
    // then answer is false
    for(int i = 1; i <= sum; i++)
        subset[0][i] = false;
 
    // Fill the subset table in bottom up manner
    for(int i = 1; i <= n; i++)
    {
        for(int j = 1; j <= sum; j++)
        {
            if (j < arr[i - 1])
                subset[i][j] = subset[i - 1][j];
 
            if (j >= arr[i - 1])
                subset[i][j] = subset[i - 1][j] ||
                subset[i - 1][j - arr[i - 1]];
        }
    }
    return subset[n][sum];
}
 
// Function to find the sum
static int findSum(int[] arr, int n)
{
    int sum = 0;
     
    // Find sum of all values
    for(int i = 0; i < n; i++)
    {
        sum += arr[i];
    }
 
    int val = (int)Math.sqrt(sum);
 
    for(int i = val; i >= 0; i--)
    {
        if (isSubsetSum(arr, n, i * i))
        {
             
            // return the value;
            return i * i;
        }
    }
    return 0;
}
 
// Driver code
public static void main(String[] args)
{
    int arr[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    int n = arr.length;
     
    System.out.println(findSum(arr, n));
}
}
 
// This code is contributed by offbeat

Python3




# Python3 program to implement
# the above approach
import math
 
def isSubsetSum(arr, n, sum):
     
    subset = [ [ True for x in range(sum + 1) ]
                      for y in range(n + 1) ]
 
    # If sum is 0, then answer is true
    for i in range(n + 1):
        subset[i][0] = True
 
    # If sum is not 0 and arr[] is empty,
    # then answer is false
    for i in range(1, sum + 1):
        subset[0][i] = False
 
    # Fill the subset table in bottom up manner
    for i in range(1, n + 1):
        for j in range(1, sum + 1):
 
            if (j < arr[i - 1]):
                subset[i][j] = subset[i - 1][j]
 
            if (j >= arr[i - 1]):
                subset[i][j] = (subset[i - 1][j] or
                                subset[i - 1][j -
                                   arr[i - 1]])
                                    
    return subset[n][sum]
 
# Function to find the sum
def findSum(arr, n):
     
    sum = 0
     
    # Find sum of all values
    for i in range(n):
        sum += arr[i]
 
    val = int(math.sqrt(sum))
 
    for i in range(val, -1, -1):
        if (isSubsetSum(arr, n, i * i)):
             
            # Return the value;
            return i * i
             
    return 0
 
# Driver Code
if __name__ == "__main__":
 
    arr = [ 1, 2, 3, 4, 5, 6, 7, 8, 9]
    n = len(arr)
     
    print(findSum(arr, n))
 
# This code is contributed by chitranayal

C#




// C# program to implement
// the above approach
using System;
 
class GFG{
 
static bool isSubsetSum(int []arr,
                        int n, int sum)
{
    bool[,] subset = new bool[n + 1, sum + 1];
 
    // If sum is 0, then answer is true
    for(int i = 0; i <= n; i++)
        subset[i, 0] = true;
 
    // If sum is not 0 and []arr is empty,
    // then answer is false
    for(int i = 1; i <= sum; i++)
        subset[0, i] = false;
 
    // Fill the subset table in bottom up manner
    for(int i = 1; i <= n; i++)
    {
        for(int j = 1; j <= sum; j++)
        {
            if (j < arr[i - 1])
                subset[i, j] = subset[i - 1, j];
 
            if (j >= arr[i - 1])
                subset[i, j] = subset[i - 1, j] ||
                subset[i - 1, j - arr[i - 1]];
        }
    }
    return subset[n, sum];
}
 
// Function to find the sum
static int findSum(int[] arr, int n)
{
    int sum = 0;
     
    // Find sum of all values
    for(int i = 0; i < n; i++)
    {
        sum += arr[i];
    }
 
    int val = (int)Math.Sqrt(sum);
 
    for(int i = val; i >= 0; i--)
    {
        if (isSubsetSum(arr, n, i * i))
        {
             
            // return the value;
            return i * i;
        }
    }
    return 0;
}
 
// Driver code
public static void Main(String[] args)
{
    int []arr = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
    int n = arr.Length;
     
    Console.WriteLine(findSum(arr, n));
}
}
 
// This code is contributed by Rohit_ranjan

Javascript




<script>
 
// JavaScript program to implement
// the above approach
 
function isSubsetSum(arr, n, sum)
{
    let subset = new Array(n + 1);
    for (var i = 0; i < subset.length; i++) {
    subset[i] = new Array(2);
    }
   
    // If sum is 0, then answer is true
    for(let i = 0; i <= n; i++)
        subset[i][0] = true;
   
    // If sum is not 0 and arr[] is empty,
    // then answer is false
    for(let i = 1; i <= sum; i++)
        subset[0][i] = false;
   
    // Fill the subset table in bottom up manner
    for(let i = 1; i <= n; i++)
    {
        for(let j = 1; j <= sum; j++)
        {
            if (j < arr[i - 1])
                subset[i][j] = subset[i - 1][j];
   
            if (j >= arr[i - 1])
                subset[i][j] = subset[i - 1][j] ||
                subset[i - 1][j - arr[i - 1]];
        }
    }
    return subset[n][sum];
}
   
// Function to find the sum
function findSum(arr, n)
{
    let sum = 0;
       
    // Find sum of all values
    for(let i = 0; i < n; i++)
    {
        sum += arr[i];
    }
   
    let val = Math.floor(Math.sqrt(sum));
   
    for(let i = val; i >= 0; i--)
    {
        if (isSubsetSum(arr, n, i * i))
        {
               
            // return the value;
            return i * i;
        }
    }
    return 0;
}
 
    // Driver Code
         
    let arr = [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ];
    let n = arr.length;
       
    document.write(findSum(arr, n));
 
</script>

Output: 

36

 

Time Complexity: O(N * SUM) 
Auxiliary Space: O(N * SUM)
 


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