"Very Similar" vs "Based On" for purpose of paper "narrative" I was trying to solve a sub-problem as part of a greater problem for which I am writing a paper to propose a solution.

I'd been working on it for quite a while, when I had the thought that it could be transformed into a special case of one of the most famous and well researched problems in computer science.

I spent about a month doing that transformation while working on other things.
All the while though, I was thinking: "Someone else must have had this idea -- its too good not to have had." I didn't spend too much time searching for it, though, because I needed an implementation and not just an idea.
I finished up the method about a week ago, and ran it, and found it to work excellently.

Today, I finally found a paper where someone did have the idea.
Almost identical to mine (I'm really not at all displeased by this)
There are, as expected, a few differences, but nothing I can't explain in a few sentences.

So the question is, in my writeup I could say:


1a. We extend the method of X, by adding ...
1b. Our method is based on the method of X, but with the following variations...
2a. The method we propose is very similar to the method of X, except ...
2b. This method is similar to that of X, but developed independently. It has the following difference.


The options 2a/b  are what actually occurred.
However, I wonder if I should not write instead 1a, or 1b because they provide a better narrative.  I've been told paper narrative is important.

I don't like the way 2b sounds, even if it is the most truthful. It sounds like I feel I have something to prove.

Is it generally worth distorting the actual order of events, for a clean narrative?
In an ordinary scientific paper, nobody expects to get a narrative of your investigation, but rather a clear explanation of how the ideas fit together.  In fact, it has been argued that success in science is precisely defined by the separation of statements from the circumstances of their creation.

I would thus argue that you should no more feel obligated to tell a reader that you invented an idea independently than that you invented it while riding a bicycle.  You may not have started by basing your method on the other method, but now that you know it exists, you can skip all sorts of argumentation and verification by changing the way you think about your method in order to base it on the prior method and the work that has been done to establish it.

Thus, I would recommend using something like your statement 1a/1b.  That is not the history of events, but based on what you report, it appears to me to be a true and more accurate characterization of the current relationship between the concepts.  Since you discovered this prior method, you are no longer proposing a new method: instead, you are applying a slight variation on an existing method, as adapted to your circumstances, and should report that fact accordingly.