Partial Permutations Comparison, Maintenance and Applications

This paper focuses on the concept of partial permutations and their use in algorithmic tasks. A partial permutation over Σ is a bijection πpar : Σ1 7→ Σ2 mapping a subset Σ1 ⊂ Σ to a subset Σ2 ⊂ Σ, where |Σ1| = |Σ2| (|Σ| denotes the size of a set Σ). Intuitively, two partial permutations agree if their mapping pairs do not form conflicts. This notion, which is formally defined in this paper, enables a consistent as well as informatively rich comparison between partial permutations. We formalize the Partial Permutations Agreement problem (PPA), as follows. Given two sets A1, A2 of partial permutations over alphabet Σ, each of size n, output all pairs (πi, πj), where πi ∈ A1, πj ∈ A2 and πi agrees with πj. The possibility of having a data structure for efficiently maintaining a dynamic set of partial permutations enabling to retrieve agreement of partial permutations is then studied, giving both negative and positive results. Applying our study enables to point out fruitful versus futile methods for efficient genes sequences comparison in database or automatic color transformation data augmentation technique for image processing through neural networks. It also shows that an efficient solution of strict Parameterized Dictionary Matching with One Gap (PDMOG) over general dictionary alphabets is not likely, unless the Strong Exponential Time Hypothesis (SETH) fails, thus negatively answering an open question posed lately.