I am reading the 1989 paper called "An O(NP) Sequence Comparison Algorithm" by Wu, Manber, Myers, and Miller. The algorithm sounds like a good fit for a project I'm doing at work. I have found some implementations in my target language that I could reuse, but I want to make sure that I understand the algorithm (and the code) because it is so crucial to my project.
I am a ways into the paper, and seem to be understanding it, but there's something important in the abstract that still doesn't make sense to me. This is the relevant portion:
Let $A$ and $B$ be two sequences of length $M$ and $N$ respectively, where without loss of generality $N \ge M$, and let $D$ be the length of a shortest edit script between them. A parameter related to $D$ is the number of deletions in such a script, $P = D/2 - (N - M)/2$. We present an algorithm...
This relationship between $P$, $D$, $N$, and $M$ is not proven in the paper. And when a try using it with a simple example (turning "xy" into "x", for instance), I get a nonsensical answer. Can someone please explain the relationship?