On Correlation Polynomials and Subword Complexity
We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2007-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3553/pdf |
| _version_ | 1827324073508601856 |
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| author | Irina Gheorghiciuc Mark Daniel Ward |
| author_facet | Irina Gheorghiciuc Mark Daniel Ward |
| author_sort | Irina Gheorghiciuc |
| collection | DOAJ |
| description | We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials. |
| first_indexed | 2024-04-25T02:04:02Z |
| format | Article |
| id | doaj.art-6ffb37e7bdc542479f4a7b318292dbca |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:04:02Z |
| publishDate | 2007-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-6ffb37e7bdc542479f4a7b318292dbca2024-03-07T14:34:52ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502007-01-01DMTCS Proceedings vol. AH,...Proceedings10.46298/dmtcs.35533553On Correlation Polynomials and Subword ComplexityIrina Gheorghiciuc0Mark Daniel Ward1Department of Mathematical SciencesDepartment of Mathematics [Philadelphia]We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials.https://dmtcs.episciences.org/3553/pdfcombinatorics on wordsaverage-case analysisautocorrelationasymptoticsanalytic methodscorrelation polynomialde \\bruijn graphdepthsubword complexitysuffix trees.[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co][info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| spellingShingle | Irina Gheorghiciuc Mark Daniel Ward On Correlation Polynomials and Subword Complexity Discrete Mathematics & Theoretical Computer Science combinatorics on words average-case analysis autocorrelation asymptotics analytic methods correlation polynomial de \\bruijn graph depth subword complexity suffix trees. [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| title | On Correlation Polynomials and Subword Complexity |
| title_full | On Correlation Polynomials and Subword Complexity |
| title_fullStr | On Correlation Polynomials and Subword Complexity |
| title_full_unstemmed | On Correlation Polynomials and Subword Complexity |
| title_short | On Correlation Polynomials and Subword Complexity |
| title_sort | on correlation polynomials and subword complexity |
| topic | combinatorics on words average-case analysis autocorrelation asymptotics analytic methods correlation polynomial de \\bruijn graph depth subword complexity suffix trees. [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| url | https://dmtcs.episciences.org/3553/pdf |
| work_keys_str_mv | AT irinagheorghiciuc oncorrelationpolynomialsandsubwordcomplexity AT markdanielward oncorrelationpolynomialsandsubwordcomplexity |