PHP Program for Subset Sum Problem

The Subset Sum Problem is a classic problem in computer science and dynamic programming. Given a set of positive integers and a target sum, the task is to determine whether there exists a subset of the given set whose elements add up to the target sum.

Using Recursive Solution

The recursive approach explores all possible combinations by either including or excluding each element ?

<?php
// A recursive solution for the subset sum problem
// Returns true if there is a subset of the set
// with a sum equal to the given sum
function isSubsetSum($set, $n, $sum)
{
   // Base Cases
   if ($sum == 0)
      return true;
   if ($n == 0 && $sum != 0)
      return false;
   
   // If the last element is greater than the sum, then ignore it
   if ($set[$n - 1] > $sum)
      return isSubsetSum($set, $n - 1, $sum);
   
   // Check if the sum can be obtained by either including or excluding the last element
   return isSubsetSum($set, $n - 1, $sum) ||
      isSubsetSum($set, $n - 1, $sum - $set[$n - 1]);
}

// Driver Code
$set = array(1, 7, 4, 9, 2);
$sum = 16;
$n = count($set);

if (isSubsetSum($set, $n, $sum) == true)
   echo "Found a subset with the given sum<br>";
else
   echo "No subset with the given sum<br>";

$sum = 25;
if (isSubsetSum($set, $n, $sum) == true)
   echo "Found a subset with the given sum.";
else
   echo "No subset with the given sum.";
?>

Found a subset with the given sum
No subset with the given sum.

In this example, the set is [1, 7, 4, 9, 2]. For sum 16, a subset [7, 9] exists. For sum 25, no valid subset is found as the maximum possible sum is 23.

Using Dynamic Programming

The dynamic programming approach builds a table to store results and avoids redundant calculations ?

<?php
// A Dynamic Programming solution for subset sum problem
// Returns true if there is a subset with sum equal to given sum
function isSubsetSum($set, $n, $sum)
{
    // The value of subset[i][j] will be true if there is a subset of
    // set[0..i-1] with sum equal to j
    $subset = array();
    
    // If sum is 0, then answer is true
    for ($i = 0; $i <= $n; $i++)
        $subset[$i][0] = true;
    
    // If sum is not 0 and set is empty, then answer is false
    for ($i = 1; $i <= $sum; $i++)
        $subset[0][$i] = false;
    
    // Fill the subset table in bottom up manner
    for ($i = 1; $i <= $n; $i++)
    {
        for ($j = 1; $j <= $sum; $j++)
        {
            if ($j < $set[$i-1])
                $subset[$i][$j] = $subset[$i-1][$j];
            else
                $subset[$i][$j] = $subset[$i-1][$j] || 
                                  $subset[$i-1][$j - $set[$i-1]];
        }
    }
    
    return $subset[$n][$sum];
}

// Driver program
$set = array(8, 15, 26, 35, 42, 59);
$sum = 50;
$n = count($set);

if (isSubsetSum($set, $n, $sum) == true)
    echo "Found a subset with given sum.";
else
    echo "No subset with given sum.";
?>

Found a subset with given sum.

In this example, the set is [8, 15, 26, 35, 42, 59] and target sum is 50. The subset [8, 42] adds up to 50, so the function returns true.

Comparison

Conclusion

The recursive solution is simple but has exponential time complexity. The dynamic programming approach is more efficient with O(n×sum) complexity, making it suitable for larger datasets and higher target sums.