Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2411.16744 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929604207312896 |
|---|---|
| author | Mathew, Martin Noda, Javier |
| author_facet | Mathew, Martin Noda, Javier |
| contents | Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This paper introduces a novel framework that presents closed-form formulas for calculating distinct permutations with replacement, fundamentally reducing the time complexity from exponential to linear relative to the sequence length for single-subword calculations. We then extend our foundational formula to handle multiple subwords through the development of an additional formula. Unlike traditional methods relying on brute-force enumeration or recursive algorithms, our approach leverages novel combinatorial constructs and advanced mathematical techniques to achieve unprecedented efficiency. This comprehensive advancement in reducing computational complexity not only simplifies permutation counting but also establishes a new benchmark for scalability and versatility. We also demonstrate the practical utility of our formulas through diverse applications, including the simultaneous identification of multiple genetic motifs in DNA sequences and complex pattern analysis in cryptographic systems, using a computer program that runs the proposed formulae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16744 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Exponential to Polynomial Complexity: Efficient Permutation Counting with Subword Constraints Mathew, Martin Noda, Javier Cryptography and Security Genomics Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This paper introduces a novel framework that presents closed-form formulas for calculating distinct permutations with replacement, fundamentally reducing the time complexity from exponential to linear relative to the sequence length for single-subword calculations. We then extend our foundational formula to handle multiple subwords through the development of an additional formula. Unlike traditional methods relying on brute-force enumeration or recursive algorithms, our approach leverages novel combinatorial constructs and advanced mathematical techniques to achieve unprecedented efficiency. This comprehensive advancement in reducing computational complexity not only simplifies permutation counting but also establishes a new benchmark for scalability and versatility. We also demonstrate the practical utility of our formulas through diverse applications, including the simultaneous identification of multiple genetic motifs in DNA sequences and complex pattern analysis in cryptographic systems, using a computer program that runs the proposed formulae. |
| title | From Exponential to Polynomial Complexity: Efficient Permutation Counting with Subword Constraints |
| topic | Cryptography and Security Genomics |
| url | https://arxiv.org/abs/2411.16744 |