Question

This is more of a puzzle than anything. I’ve actually found a solution but it is so slow I thought I lost my internet connection (see below).

Here’s the problem:

Let’s say I have an array of numbers, like so:

$numbers_array = array(1, 2, 3, 4, 5, 6, 7, 8, 9);

Let’s also say that I have a number some numbers, stored in variables like so:

$sum = 15;
$sum2 = 24;
$sum3 = 400;

I am trying to create a function that will return true if any of the numbers in $numbers_array can be added together (each only used once) to form the sums:

function is_summable($array_of_nums, $sum_to_check) {
    //What to put here?
}

var_dump(is_summable($numbers_array, $sum));
var_dump(is_summable($numbers_array, $sum2));
var_dump(is_summable($numbers_array, $sum3));

The above should output:

bool(true)
bool(true)
bool(false)

Because 7 + 8 = 15, 7 + 8 + 9 = 24, but no combination of 1-9 can create 200.

Here’s my EXTREMELY slow solution:

function is_summable($numbers, $sum) {
    //Sort provided numbers and assign numerical keys.
    asort($numbers);
    $numbers = array_values($numbers);

    //Var for additions and var for number of provided numbers.
    $total = 0;
    $numbers_length = count($numbers);

    //Empty var to fill below.
    $code = '';

    //Loop and add for() loops.
    for ($i = 0; $i < $numbers_length; $i++) {
        $code .= 'for ($n' . $i . ' = 0; $n' . $i . ' < ' . $numbers_length . '; $n' . $i . '++) {';

        if ($i != 0) {
            $code .= 'if ($n' . $i . ' != $n' . ($i - 1) . ') {';
        }

        $code .= '$total += intval($numbers[$n' . $i . ']);';
        $code .= 'if ($total == $sum) {';
        $code .= 'return true;';
        $code .= '}';
    }

    //Add ending bracket for for() loops above.
    for ($l = 0; $l < $numbers_length; $l++) {
        $code .= '$total -= intval($numbers[$n' . $i . ']);';
        if ($l != 0) {
            $code .= '}';
        }
        $code .= '}';
    }

    //Finally, eval the code.
    eval($code);

    //If "true" not returned above, return false.
    return false;
}

$num_arr = array(1,2,3,4,5,6,7,8,9);
var_dump(is_summable($num_arr, 24));

http://pastebin.com/1nawuwXK

As always, help is appreciated!!

Answer 1

Your problem is in fact a standard algorithmic problem (as Jon mentioned knapsack problem), more specifically Subset sum problem. It can be solved in polynomial time (have look on wiki page).

Pseudocode:

initialize a list S to contain one element 0.
for each i from 1 to N do
  let T be a list consisting of xi + y, for all y in S
  let U be the union of T and S
  sort U
  make S empty 
  let y be the smallest element of U 
  add y to S 
  for each element z of U in increasing order do
     //trim the list by eliminating numbers close to one another
     //and throw out elements greater than s
    if y + cs/N < z ≤ s, set y = z and add z to S 
if S contains a number between (1 − c)s and s, output yes, otherwise no