Preface It's not uncommon for a written piece of code to be both brief and functionality correct, yet difficult to reason about. This is especially true of recursive algorithms, which can require...
Preface
It's not uncommon for a written piece of code to be both brief and functionality correct, yet difficult to reason about. This is especially true of recursive algorithms, which can require some amount of simulating the algorithm mentally (or on a whiteboard) on smaller problems to try to understand the underlying logic. The more you have to perform these manual simulations, the more difficult it becomes to track what exactly is going on at any stage of computation. It's also not uncommon that these algorithms can be made easier to reason about with relatively small changes, particularly in the way you conceptualize the solution to the problem. Our goal will be to take a brief tour into what these changes might look like and why they are effective at reducing our mental overhead.
Background
We will consider the case of the subset sum problem, which is essentially a special case of the knapsack problem where you have a finite number of each item and each item's value is equal to its weight. In short, the problem is summarized as one of the following:

Given a set of numbers, is there a subset whose sum is exactly equal to some target value?

Given a set of numbers, what is the subset whose sum is the closest to some target value without exceeding it?
For example, given the set of numbers {1, 3, 3, 5}
and a target value of 9
, the answer for both of those questions is {1, 3, 5}
because the sum of those numbers is 9
. For a target value of 10
, however, the first question has no solution because no combination of numbers in the set {1, 3, 3, 5}
produces a total of 10
, but the second question produces a solution of {1, 3, 5}
because 9
is the closest value to 10
that those numbers can produce without going over.
A Greedy Example
We'll stick to the much simpler case of finding an exact match to our target value so we don't have to track what the highest value found so far is. To make things even simpler, we'll consider the case where all numbers are positive, nonzero integers. This problem can be solved with some naive recursionsimply try all combinations until either a solution is found or all combinations have been exhausted. While more efficient solutions exist, naive recursion is the easiest to conceptualize.
An initial assessment of the problem seems simple enough. Our solution is defined as the set of array elements whose total is equal to our target value. To achieve this, we loop through each of the elements in the array, try combinations with all of the remaining elements, and keep track of what the current total is so we can compare it to our target. If we find an exact match, we return an array containing the matching elements, otherwise we return nothing. This gives us something like the following:
function subsetSum($target_sum, $values, $total = 0) {
// Base case: a total exceeding our target sum is a failure.
if($total > $target_sum) {
return null;
}
// Base case: a total matching our target sum means we've found a match.
if($total == $target_sum) {
return array();
}
foreach($values as $index=>$value) {
// Recursive case: try combining the current array element with the remaining elements.
$result = subsetSum($target_sum, array_slice($values, $index + 1), $total + $value);
if(!is_null($result)) {
return array_merge(array($value), $result);
}
}
return null;
}
Your Scope is Leaking
This solution works. It's functionally correct and will produce a valid result every single time. From a purely functional perspective, nothing is wrong with it at all; however, it's not easy to follow what's going on despite how short the code is. If we look closely, we can tell that there are a few major problems:

It's not obvious at first glance whether or not the programmer is expected to provide the third argument. While a default value is provided, it's not clear if this value is only a default that should be overridden or if the value should be left untouched. This ambiguity means relying on documentation to explain the intention of the third argument, which may still be ignored by an inattentive developer.

The base case where a failure occurs, i.e. when the accumulated total exceeds the target sum, occurs one stack frame further into the recursion than when the total has been incremented. This forces us to consider not only the current iteration of recursion, but one additional iteration deeper in order to track the flow of execution. Ideally an iteration of recursion should be conceptually isolated from any other, limiting our mental scope to only the current iteration.

We're propagating an accumulating total that starts from
0
and increments toward our target value, forcing us to to track two different values simultaneously. Ideally we would only track one value if possible. If we can manage that, then the ambiguity of the third argument will be eliminated along with the argument itself.
Overall, the amount of code that the programmer needs to look at and the amount of branching they need to follow manually is excessive. The function is only 22 lines long, including whitespace and comments, and yet the amount of effort it takes to ensure you're understanding the flow of execution correctly is pretty significant. This is a pretty good indicator that we probably did something wrong. Something so simple and short shouldn't take so much effort to understand.
Patching the Leak
Now that we've assessed the problems, we can see that our original solution isn't going to cut it. We have a couple of ways we could approach fixing our function: we can either attempt to translate the abstract problems into tangible solutions or we can modify the way we've conceptualized the solution. With that in mind, let's take a second crack at this problem by trying the latter.
We've tried taking a look at this problem from a topdown perspective: "given a target value, are there any elements that produce a sum exactly equal to it?" Clearly this perspective failed us. Instead, let's try flipping the equation: "given an array element, can it be summed with others to produce the target value?"
This fundamentally changes the way we can think about the problem. Previously we were hung up on the idea of keeping track of the current total sum of the elements we've encountered so far, but that approach is incompatible with the way we're thinking of this problem now. Rather than incrementing a total, we now find ourselves having to do something entirely different: if we want to know if a given array element is part of the solution, we need to first subtract the element from the problem and find out if the smaller problem has a solution. That is, to find if the element 3
is part of the solution for the target sum of 8
, then we're really asking if 3 + solutionFor(5)
is valid.
The new solution therefore involves looping over our array elements just as before, but this time we check if there is a solution for the target sum minus the current array element:
function subsetSum($target_sum, $values) {
// Base case: the solution to the target sum of 0 is the empty set.
if($target_sum === 0) {
return array();
}
foreach($values as $index=>$value) {
// Base case: any element larger than our target sum cannot be part of the solution.
if($value > $target_sum) {
continue;
}
// Recursive case: do the remaining elements create a solution for the subproblem?
$result = subsetSum($target_sum  $value, array_slice($values, $index + 1));
if(!is_null($result)) {
return array_merge(array($value), $result);
}
}
return null;
}
A Brief Review
With the changes now in place, let's compare our two functions and, more importantly, compare our new function to the problems we assessed with the original. A few brief points:

Both functions are the same exact length, being only 22 lines long with the same number of comments and an identical amount of whitespace.

Both functions touch the same number of elements and produce the same output given the same input. Apart from a change in execution order of a base case, functionality is nearly identical.

The new function no longer requires thinking about the scope of next iteration of recursion to determine whether or not an array element is included in the result set. The base case for exceeding the target sum now occurs prior to recursion, keeping the scope of the value comparison nearest where those values are defined.

The new function no longer uses a third accumulator argument, reducing the number of values to be tracked and removing the issue of ambiguity with whether or not to include the third argument in toplevel calls.

The new function is now defined in terms of finding the solutions to increasingly smaller target sums, making it easier to determine functional correctness.
Considering all of the above, we can confidently state that the second function is easier to follow, easier to verify functional correctness for, and less confusing for anyone who needs to use it. Although the two functions are nearly identical, the second version is clearly and objectively better than the original. This is because despite both being functionally correct, the first function does a poor job at accurately defining the problem it's solving while the second function is clear and accurate in its definition.
Correct code isn't necessarily accurate code. Anyone can write code that works, but writing code that accurately defines a problem can mean the difference between understanding what you're looking at, and being completely bewildered at how, or even why, your code works in the first place.
Final Thoughts
Accurately defining a problem in code isn't easy. Sometimes you'll get it right, but more often than not you'll get it wrong on the first go, and it's only after you've had some distance from you original solution that you realize that you should've done things differently. Despite that, understanding the difference between functional correctness and accuracy gives you the opportunity to watch for obvious inaccuracies and keep them to a minimum.
In the end, even functionally correct, inaccurate code is worth more than no code at all. No amount of theory is a replacement for practical experience. The only way to get better is to mess up, assess why you messed up, and make things just a little bit better the next time around. Theory just makes that a little easier.