Parameterized Matching

Problem Definition

Parameterized strings, or p-strings, are strings that contain both ordinary symbols from an alphabet Σ and parameter symbols from an alphabet Π. Two equal-length p‑strings s and ${ s^{\prime} }$ are a parameterized match, or p-match, if one p‑string can be transformed into the other by applying a one-to-one function that renames the parameter symbols. The following example of a p-match is one with both ordinary and parameter symbols. The ordinary symbols are in lowercase and the parameter symbols are in uppercase.

$$ \begin{aligned} s &= A\ b\ A\ b\ C\ A\ d\ b\ A\ C\ d\ d \\ s^{\prime} &= D\ b\ D\ b\ E\ D\ d\ b\ D\ E\ d\ d \end{aligned} $$

In some of the problems to be considered it will be sufficient to solve for p-strings in which all symbols are parameter symbols, as this is the more difficult part of the problem. In other words, the case in which ${ \Sigma = \emptyset }$. In this case the definition can be reformulated so that s and ${ s^{\prime} }$...