Recursively-generated Iterators

[Roy Johnson]

Recursive algorithms are often simple and intuitive. Unfortunately, they are also often explosive in terms of memory and execution time required. Take, for example, the N-choose-M algorithm:

# Given a list of M items and a number N,
# generate all size-N subsets of M
sub choose_n {
    my $n = pop;
    # Base cases
    return []   if $n == 0  or $n > @_;
    return [@_] if $n == @_;
    # otherwise..
    my ($first, @rest) = @_;
    # combine $first with all N-1 combinations of @rest,
    # and generate all N-sized combinations of @rest
    my @include_combos = choose_n(@rest, $n-1);
    my @exclude_combos = choose_n(@rest, $n);
    return ( (map {[$first, @$_]} @include_combos)
         , @exclude_combos );
}

Great, as long as you don't want to generate all 10-element subsets of a 20-item list. Or 45-choose-20. In those cases, you will need an iterator. Unfortunately, iteration algorithms are generally completely unlike the recursive ones they mimic. They tend to be a lot trickier.

But they don't have to be. You can often write iterators that look like their recursive counterparts — they even include recursive calls — but they don't suffer from explosive growth. That is, they'll still take a long time to get through a billion combinations, but they'll start returning them to you right away, and they won't eat up all your memory.

The trick is to create iterators to use in place of your recursive calls, then do a little just-in-time placement of those iterator creations.

So let's take a first stab at choose_n. First, our base cases are going to be subs that return whatever they were returning before, but after returning those values once, they don't return anything anymore:

sub iter_choose_n {
    my $n = pop;
    # Base cases
    my $once = 0;
    return sub {$once++ ? () : []} if $n == 0  or $n > @_;
    my ($first, @rest) = @_;
    return sub {$once++ ? () : [$first, @rest]} if $n == @_;

Apart from the iterator trappings, we've got essentially what we had before. Converting the map into an iterator involves some similar work, but the parallels are still pretty obvious. We exhaust the first iterator before turning to the second:

    # otherwise..
    my $include_iter = iter_choose_n(@rest, $n-1);
    my $exclude_iter = iter_choose_n(@rest, $n);
    return sub {
        if (my $set = $include_iter->()) {
            return [$first, @$set];
        }
        else {
            return $exclude_iter->();
        }
    }

We now have a recursively-defined iterator that wasn't a heck of a lot more complex than our original algorithm. That's the good news. The bad news is: it's still doubly recursive, O(2^N) in space and time, and so will take a long time to start generating data. Time for a little trick. Because we don't use $exclude_iter until we've exhausted $include_iter, we can delay defining it:

    # otherwise..
    my $include_iter = iter_choose_n(@rest, $n-1);
    my $exclude_iter;
    return sub {
        if (my $set = $include_iter->()) {
            return [$first, @$set];
        }
        else {
            $exclude_iter ||= iter_choose_n(@rest, $n);
            return $exclude_iter->();
        }
    }
}

Now our code is singly recursive, O(N) in space and time to generate an iterator, and that makes a big difference. Big enough that you probably won't need to go to the trouble of coming up with an O(1) truly iterative solution.

Of course, if you complete the iterations, eventually you will have generated those 2^N subs, and they'll clog up your memory. You may not be concerned about that (you may not be expecting to perform all that many iterations), but if you are, you can put a little code in to free up exhausted iterators:

    # otherwise..
    my $include_iter = iter_choose_n(@rest, $n-1);
    my $exclude_iter;
    return sub {
        if ($include_iter and my $set = $include_iter->()) {
            return [$first, @$set];
        }
        else {
            if ($include_iter) {
                undef $include_iter;
                $exclude_iter = iter_choose_n(@rest, $n);
            }
            return $exclude_iter->();
        }
    }
}

Update: See my response to Kelan for a recipe for conversion from recursion to iterator.