Find combinations of numbers that sum to some desired number

Question

I need an algorithm that identifies all possible combinations of a set of numbers that sum to some other number.

For example, given the set {2,3,4,7}, I need to know all possible subsets that sum to x. If x == 12, the answer is {2,3,7}; if x ==7 the answer is {{3,4},{7}} (ie, two possible answers); and if x==8 there is no answer. Note that, as these example imply, numbers in the set cannot be reused.

This question was asked on this site a couple years ago but the answer is in C# and I need to do it in Perl and don't know enough to translate the answer.

I know that this problem is hard (see other post for discussion), but I just need a brute-force solution because I am dealing with fairly small sets.

Answer 1

sub Solve
{
  my ($goal, $elements) = @_;

  # For extra speed, you can remove this next line
  # if @$elements is guaranteed to be already sorted:
  $elements = [ sort { $a <=> $b } @$elements ];

  my (@results, $RecursiveSolve, $nextValue);

  $RecursiveSolve = sub {
    my ($currentGoal, $included, $index) = @_;

    for ( ; $index < @$elements; ++$index) {
      $nextValue = $elements->[$index];
      # Since elements are sorted, there's no point in trying a
      # non-final element unless it's less than goal/2:
      if ($currentGoal > 2 * $nextValue) {
        $RecursiveSolve->($currentGoal - $nextValue,
                          [ @$included, $nextValue ],
                          $index + 1);
      } else {
        push @results, [ @$included, $nextValue ]
            if $currentGoal == $nextValue;
        return if $nextValue >= $currentGoal;
      }
    } # end for
  }; # end $RecursiveSolve

  $RecursiveSolve->($goal, [], 0);
  undef $RecursiveSolve; # Avoid memory leak from circular reference
  return @results;
} # end Solve


my @results = Solve(7, [2,3,4,7]);
print "@$_\n" for @results;

This started as a fairly direct translation of the C# version from the question you linked, but I simplified it a bit (and now a bit more, and also removed some unnecessary variable allocations, added some optimizations based on the list of elements being sorted, and rearranged the conditions to be slightly more efficient).

I've also now added another significant optimization. When considering whether to try using an element that doesn't complete the sum, there's no point if the element is greater than or equal to half the current goal. (The next number we add will be even bigger.) Depending on the set you're trying, this can short-circuit quite a bit more. (You could also try adding the next element instead of multiplying by 2, but then you have to worry about running off the end of the list.)

Answer 2

The rough algorithm is as follows:

have a "solve" function that takes in a list of numbers already included and a list of those not yet included.

• This function will loop through all the numbers not yet included.
• If adding that number in hits the goal then record that set of numbers and move on,
• if it is less than the target recursively call the function with the included/exluded lists modified with the number you are looking at.
• else just go to the next step in the loop (since if you are over there is no point trying to add more numbers unless you allow negative ones)
• You call this function initially with your included list empty and your yet to be included list with your full list of numbers.

There are optimisations you can do with this such as passing the sum around rather than recalculating each time. Also if you sort your list initially you can do optimisations based on the fact that if adding number k in the list makes you go over target then adding k+1 will also send you over target.

Hopefully that will give you a good enough start. My perl is unfortuantely quite rusty.

Pretty much though this is a brute force algorithm with a few shortcuts in it so its never going to be that efficient.