Maximum subsequence sum from a given array which is a perfect square
Last Updated :
26 Oct, 2023
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 * 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++
#include <bits/stdc++.h>
using namespace std;
bool isSubsetSum( int arr[], int n, int sum)
{
bool subset[n + 1][sum + 1];
for ( int i = 0; i <= n; i++)
subset[i][0] = true ;
for ( int i = 1; i <= sum; i++)
subset[0][i] = false ;
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];
}
int findSum( int * arr, int n)
{
int sum = 0;
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 i * i;
}
}
return 0;
}
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
import java.util.*;
class GFG{
static boolean isSubsetSum( int arr[],
int n, int sum)
{
boolean [][] subset = new boolean [n + 1 ][sum + 1 ];
for ( int i = 0 ; i <= n; i++)
subset[i][ 0 ] = true ;
for ( int i = 1 ; i <= sum; i++)
subset[ 0 ][i] = false ;
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];
}
static int findSum( int [] arr, int n)
{
int sum = 0 ;
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 i * i;
}
}
return 0 ;
}
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));
}
}
|
Python3
import math
def isSubsetSum(arr, n, sum ):
subset = [ [ True for x in range ( sum + 1 ) ]
for y in range (n + 1 ) ]
for i in range (n + 1 ):
subset[i][ 0 ] = True
for i in range ( 1 , sum + 1 ):
subset[ 0 ][i] = False
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 ]
def findSum(arr, n):
sum = 0
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 i * i
return 0
if __name__ = = "__main__" :
arr = [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ]
n = len (arr)
print (findSum(arr, n))
|
C#
using System;
class GFG{
static bool isSubsetSum( int []arr,
int n, int sum)
{
bool [,] subset = new bool [n + 1, sum + 1];
for ( int i = 0; i <= n; i++)
subset[i, 0] = true ;
for ( int i = 1; i <= sum; i++)
subset[0, i] = false ;
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];
}
static int findSum( int [] arr, int n)
{
int sum = 0;
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 i * i;
}
}
return 0;
}
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));
}
}
|
Javascript
<script>
function isSubsetSum(arr, n, sum)
{
let subset = new Array(n + 1);
for ( var i = 0; i < subset.length; i++) {
subset[i] = new Array(2);
}
for (let i = 0; i <= n; i++)
subset[i][0] = true ;
for (let i = 1; i <= sum; i++)
subset[0][i] = false ;
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 findSum(arr, n)
{
let sum = 0;
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 i * i;
}
}
return 0;
}
let arr = [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ];
let n = arr.length;
document.write(findSum(arr, n));
</script>
|
Time Complexity: O(N * SUM)
Auxiliary Space: O(N * SUM)
Efficient approach : Space optimization
In previous approach the current value subset[i][j] is only depend upon the current and previous row values of DP. So to optimize the space complexity we use a single 1D array to store the computations.
Implementation steps:
- Create a 1D array subset of size sum+1 and initialize it with false.
- Set a base case and initialize subset[0] = True.
- Now iterate over subproblems by the help of nested loop and get the current value from previous computations.
- At last return the final answer stored in subset[sum].
Implementation:
C++
#include <bits/stdc++.h>
using namespace std;
bool isSubsetSum( int arr[], int n, int sum)
{
bool subset[sum + 1];
memset (subset, false , sizeof (subset));
subset[0] = true ;
for ( int i = 0; i < n; i++) {
for ( int j = sum; j >= arr[i]; j--) {
subset[j] = subset[j] || subset[j - arr[i]];
}
}
return subset[sum];
}
int findSum( int * arr, int n)
{
int sum = 0;
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 i * i;
}
}
return 0;
}
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
import java.io.*;
import java.util.*;
class GFG {
static boolean isSubsetSum( int [] arr, int n, int sum)
{
boolean [] subset = new boolean [sum + 1 ];
Arrays.fill(subset, false );
subset[ 0 ] = true ;
for ( int i = 0 ; i < n; i++) {
for ( int j = sum; j >= arr[i]; j--) {
subset[j] = subset[j] || subset[j - arr[i]];
}
}
return subset[sum];
}
static int findSum( int [] arr, int n)
{
int sum = 0 ;
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 i * i;
}
}
return 0 ;
}
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));
}
}
|
Python3
import math
def isSubsetSum(arr, n, target):
subset = [ False ] * (target + 1 )
subset[ 0 ] = True
for i in range (n):
for j in range (target, arr[i] - 1 , - 1 ):
subset[j] = subset[j] or subset[j - arr[i]]
return subset[target]
def findSum(arr, n):
total_sum = sum (arr)
val = int (math.sqrt(total_sum))
for i in range (val, - 1 , - 1 ):
if isSubsetSum(arr, n, i * i):
return i * i
return 0
if __name__ = = "__main__" :
arr = [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ]
n = len (arr)
print (findSum(arr, n))
|
C#
using System;
public class GFG {
static bool isSubsetSum( int [] arr, int n, int sum)
{
bool [] subset = new bool [sum + 1];
Array.Fill(subset, false );
subset[0] = true ;
for ( int i = 0; i < n; i++) {
for ( int j = sum; j >= arr[i]; j--) {
subset[j] = subset[j] || subset[j - arr[i]];
}
}
return subset[sum];
}
static int findSum( int [] arr, int n)
{
int sum = 0;
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 i * i;
}
}
return 0;
}
public static void Main()
{
int [] arr = { 1, 2, 3, 4, 5, 6, 7, 8, 9 };
int n = arr.Length;
Console.WriteLine(findSum(arr, n));
}
}
|
Javascript
function isSubsetSum(arr, n, sum) {
const subset = new Array(sum + 1).fill( false );
subset[0] = true ;
for (let i = 0; i < n; i++) {
for (let j = sum; j >= arr[i]; j--) {
subset[j] = subset[j] || subset[j - arr[i]];
}
}
return subset[sum];
}
function findSum(arr, n) {
let sum = 0;
for (let i = 0; i < n; i++) {
sum += arr[i];
}
const val = Math.floor(Math.sqrt(sum));
for (let i = val; i >= 0; i--) {
if (isSubsetSum(arr, n, i * i)) {
return i * i;
}
}
return 0;
}
const arr = [1, 2, 3, 4, 5, 6, 7, 8, 9];
const n = arr.length;
console.log(findSum(arr, n));
|
Time Complexity: O(N * SUM)
Auxiliary Space: O(SUM)
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