String distance problems typically ask for a minimum number of permitted operations to transform one string into another. Such problems find application in a wide variety of areas, including error-correcting codes, parsing theory, speech recognition, and computational biology, to name a few. Here we consider a classic string distance problem, the NPcomplete String-to-String Correction problem, first studied by Wagner some 35 years ago. In this problem, we are asked whether it is possible to transform string x into string y with at most k operations on x, where permitted operations are single-character deletions and adjacent character exchanges. We prove that String-to-String Correction is fixedparameter tractable, for parameter k, and present a simple fixed-parameter algorithm that solves the problem in O(2kn) time. We also devise a bounded search tree algorithm, and introduce a bookkeeping technique that we call charge and reduce. This leads to an algorithm whose running time is O(1.6181kn).
Abu Khzam, F., Fernau, H., Langston, M., Lee-Cultura, S., & Stege, U. (2011). Charge and reduce: A fixed-parameter algorithm for String-to-String Correction. Discrete Optimization, 8(1), 41-49. https://doi.org/10.1016/j.disopt.2010.10.003