bemo bemo - 3 years ago 145
PHP Question

Find all combinations of x numbers where they sum to Y

I'm trying to write this solution in PHP.


number of numbers = x
smallest number = 1
sum of numbers = y

I'm not dealing with very large numbers, largest x is approximatly 50, largest y is approximatly 80.

rules: Within each set of numbers, the number proceeding the previous must be equal to or greater.

For example

x = 3
min = 1
y = 6


note that (3,2,1) isn't a solution as they are in descending order.

Answer Source

This is easily solved via recursion. The time complexity though will be high. For a better (but slightly more complex solution) use dynamic programming.

Here's the idea:

If the size of the set is 1 then the only possible solution is the desired sum.

If the set is larger than one then you can merge a number X between the minimum and the desired sum with a set of numbers which add up to the desired sum minus X.

function tuplesThatSumUpTo($desiredSum, $minimumNumber, $setSize) {
      $tuples = [];  
      if ($setSize <= 1) {
          return [ [ $desiredSum ] ]; //A set of sets of size 1 e.g. a set of the desired sum
      for ($i = $minimumNumber;$i < $desiredSum;$i++) {
          $partial = tuplesThatSumUpTo($desiredSum-$i, $minimumNumber,$setSize-1); 
          $tuples = array_merge($tuples, array_map(function ($tuple) use ($i) {
              $res = array_merge([$i], $tuple);          
              return $res;
      return array_unique($tuples,SORT_REGULAR);

See it run:

The dynamic programming approach would have you instead hold an array of sets with partial sums and refer back to it to fill in what you need later on.

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