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

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: 9 

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 * 2^N) 
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)