Partial permutations comparison, maintenance and applications

Abstract

This paper studies partial permutations and their use in algorithmic tasks. A partial permutation over Σ is a bijection ${\pi }_{par}:{\Sigma }_1\mapsto {\Sigma }_2$ mapping a subset ${\Sigma }_1\subset \Sigma$ to a subset ${\Sigma }_2\subset \Sigma$, where $\left|{\Sigma }_1\right|=\left|{\Sigma }_2\right|$ ($\left|\Sigma \right|$ denotes the size of a set Σ). Intuitively, two partial permutations agree if their mapping pairs do not form conflicts. We formally define this notion enabling a consistent as well as informatively rich comparison between partial permutations. We define the Partial Permutations Agreement problem (PPA), as follows. Given two sets $A_1,A_2$ of partial permutations over alphabet Σ, each of size n, output a pair (${\pi }_i,{\pi }_j)$, where ${\pi }_i\in A_1,{\pi }_j\in A_2$ and ${\pi }_i$ agrees with ${\pi }_j$, if exists. We study the existence of a data structure for efficiently maintaining a dynamic set of partial permutations enabling to retrieve agreement of partial permutations giving both negative and positive results. As applications we point out: (1) fruitful/futile methods for efficient genes sequences comparison in database, (2) an automatic color transformation data augmentation technique for image processing through neural networks, (3) negatively answer a recently posed open question on the strict parameterized dictionary matching with one gap (PDMOG) problem over general dictionary alphabets.