The Two Sum Problem Explained

algorithms, math

Just over three years ago, I watched this video that goes over the so-called “Two Sum” problem for the first time. The problem statement is as follows:

Given a sorted list of integers (unimaginitively called numbers), determine if any 2 integers in the list sum up to a number N.

To be honest I did not understand why the proposed optimal solution that uses 2 pointers works the way it does, without missing any possible pairs. The explanation given by the interview candidate in the video always struck me as way too hand-wavy for my tastes.

And to be really honest I never bothered to convince myself that the 2-pointer approach is correct. Until today. This post is about the correctness behind the 2-pointer method, because I have yet to see a clear explanation about this topic.

Brute force approach

First let’s look at the brute-force solution. The brute-force solution looks at every single possible pair of numbers by using a double-for-loop. This is a very common pattern (nested looping) whenever one wants to consider all possible combinations, where each for-loop corresponds to a single “dimension” we want to exhaustively pore over. In this case there are 2 dimensions because there are 2 numbers we need to look at, so we must use 2 for-loops.

Here is the basic pseudocode 1:

for i in numbers:
  for j in numbers:
    if i + j == N:
      return i, j

I think even beginner programmers can see that the brute force approach works. We just look at every possible 2-number combination to see if they will add up to N, and if so, we stop the search. Simple! 2

2-pointer method

The 2-pointer method boils down to the following observation:

Remove numbers from the pairwise search if they cannot be used (with any other number) to sum up to N.

Although the solution posted in countless places online involve pointers, it is more intuitive to think of modifying the list after each pairwise inspection. Below is the algorithm in plain English:

  1. Construct a pair of numbers (a, b) such that a is the smallest number and b is the biggest number in the list. That is, these are the leftmost and rightmost ends of the sorted list, respectively.
  2. If the sum of a + b is equal to N, of course we’re done.
  3. If the sum of a + b is bigger than N, delete b from the list. Go back to Step 1.
  4. If the sum of a + b is smaller than N, delete a from the list. Go back to Step 1.
  5. If the list becomes smaller than 2 elements, stop (obviously, because there are no more pairs to consider). Optionally return an error.

The algorithm described above can be equivalently coded with pointers, so there is no material difference to discuss in terms of implementation.

Anyway, we just need to make sense of the critical Steps, namely Steps 3, 4, and 5, and that should be enough to quell any worries about correctness.

Step 3

This is the step that removes the largest element b in the list from consideration for all future iterations. How can this be correct?

Well, let’s consider an example. If N is 50 but a + b is 85, we must look for a smaller sum. This much is obvious.

We just used a and b to get to 85, but because we must get to a smaller sum, we would like to swap out either a or b (or both, eventually) with another number from the list. The question is, which one do we swap out?

We can’t replace a with the next bigger number (or any other number between a and b), because doing so will result in a sum that is at least as big as 85 (or bigger). So a has to stay — we can’t rule out other combinations of a with some number other than b (maybe a and its next-biggest neighbor, etc).

That leaves us with b. We throw out b and replace it with the next biggest number, which is guaranteed to be less than or equal to the just-thrown-out b, because the list is sorted.

In other words, all pairs of b and every other element in the list already sums up to 85 or some other higher number. So b is a red herring that’s leading us astray. We must throw it out.

Step 4

This is the “mirror” of Step 3. Here we throw out the smallest number out of future pairwise searches, because we know that a, no matter which number it is paired with (even with the biggest one, b), is too small to meet the target N. In other words, a fails to give enough of a “boost” to any other number to reach N. It is very much useless to the remaining other candidates, and so we throw it out.

Step 5

This Step’s analogy when using pointers is to consider the condition when the pointers “cross”. The pointers “crossing”, in and of itself, doesn’t seem particularly significant. However when we view this condition by looking at the dwindling size of the overall list (by chopping off the left and right ends in Steps 4 and 3), the point becomes obvious. We must stop when the list becomes too small to make Step 1 impossible to fulfill (namely, the construction of the pair (a, b)), due to the fact that there aren’t enough elements in the list (singleton or empty list).

2-pointer method, in pseudocode

For sake of completeness, here is the pseudocode for the same algorithm. You will see how using pointers (instead of deleting elements outright as described in Steps 3 and 4) doesn’t change the algorithm at all.

# Partial implementation of Step 5. Early exit if list is too small to begin with.
if length(numbers) < 2:
  return error

# Step 1.
a_idx = 0
b_idx = length(numbers) - 1
sum = numbers[a_idx] + numbers[b_idx]

# Begin search, but only if we have to search.
while sum != N:
  # Step 3
  if sum > N:
    b_idx -= 1
  # Step 4
  elif sum < N:
    a_idx += 1

  # Step 5
  if a_idx == b_idx:
    return error

  # Step 1 (again, because we didn't find a match above).
  sum = numbers[a_idx] + numbers[b_idx]

# Step 2
return numbers[a_idx], numbers[b_idx]

It may be of interest to readers who are fairly new to programming that Step 2 comes in at the very end. Getting the “feel” for converting plain-English algorithms into actual code is something that requires experience, and can only be acquired with practice over time.

Do the pointers ever skip over each other?

It is worth pointing out that the condition a_idx == b_idx is well-formed. That is, there will never be a case where a_idx and b_idx will somehow “skip over” each other, rendering the if-condition useless. This is because we only ever increment a_idx or decrement b_idx, exclusively — that is, we never modify both of them within the same iteration. So, the variables only ever change by ±1, and at some point, if the search goes on long enough, the indices are bound to converge at the same numerical value.


I think the beauty of this problem is that it’s so simple, and yet it is also a very cool way of looking at the problem of search. Steps 3 and 4 are essentially very aggressive (and correct!) eliminations of bad search paths. There’s just something refreshing about eliminating entire branches of a search tree to speed things up.

If you compare the 2-pointer method with the brute force approach, it is in essence doing the same logical thing, with fewer steps. Whereas the brute force approach performs a pairwise comparison across all possible combinations, the 2-pointer method preemptively discards many combinations by removing elements outright from future consideration. That’s the kind of power you need to go from \(O(n^2)\) to \(O(n)\)!

Hope this helps!

  1. Of course, this pseudocode ignores edge-cases, but I didn’t want to clutter the code listing with non-essential ideas.↩︎

  2. As an added benefit, the brute-force approach works even if the input list is not sorted.↩︎