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Partial Permutations Comparison, Maintenance and Applications

Authors Avivit Levy , Ely Porat, B. Riva Shalom

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Author Details

Avivit Levy
  • Department of Software Engineering, Shenkar College, Ramat-Gan, Israel
Ely Porat
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
B. Riva Shalom
  • Department of Software Engineering, Shenkar College, Ramat-Gan, Israel


We would like to thank Amihood Amir for his valuable suggestions while reading a former version of the paper.

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Avivit Levy, Ely Porat, and B. Riva Shalom. Partial Permutations Comparison, Maintenance and Applications. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 10:1-10:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


This paper focuses on the concept of partial permutations and their use in algorithmic tasks. A partial permutation over Σ is a bijection π_{par}: Σ₁↦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 (π_i, π_j), where π_i ∈ A₁, π_j ∈ A₂ and π_i agrees with π_j. 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Partial permutations
  • Partial words
  • Genes comparison
  • Color transformation
  • Dictionary matching with gaps
  • Parameterized matching
  • SETH hypothesis


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