String Indexing for Top-$k$ Close Consecutive Occurrences
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866909105381179392 |
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| author | Bille, Philip Gørtz, Inge Li Pedersen, Max Rishøj Rotenberg, Eva Steiner, Teresa Anna |
| author_facet | Bille, Philip Gørtz, Inge Li Pedersen, Max Rishøj Rotenberg, Eva Steiner, Teresa Anna |
| contents | The classic string indexing problem is to preprocess a string $S$ into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string $P$, report all occurrences of $P$ within $S$. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-$k$ close consecutive occurrences problem (SITCCO). Here, a consecutive occurrence is a pair $(i,j)$, $i < j$, such that $P$ occurs at positions $i$ and $j$ in $S$ and there is no occurrence of $P$ between $i$ and $j$, and their distance is defined as $j-i$. Given a pattern $P$ and a parameter $k$, the goal is to report the top-$k$ consecutive occurrences of $P$ in $S$ of minimal distance. The challenge is to compactly represent $S$ while supporting queries in time close to the length of $P$ and $k$. We give three time-space trade-offs for the problem. Let $n$ be the length of $S$, $m$ the length of $P$, and $ε\in(0,1]$. Our first result achieves $O(n\log n)$ space and optimal query time of $O(m+k)$. Our second and third results achieve linear space and query times either $O(m+k^{1+ε})$ or $O(m + k \log^{1+ε} n)$. Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_04128 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | String Indexing for Top-$k$ Close Consecutive Occurrences Bille, Philip Gørtz, Inge Li Pedersen, Max Rishøj Rotenberg, Eva Steiner, Teresa Anna Data Structures and Algorithms The classic string indexing problem is to preprocess a string $S$ into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string $P$, report all occurrences of $P$ within $S$. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-$k$ close consecutive occurrences problem (SITCCO). Here, a consecutive occurrence is a pair $(i,j)$, $i < j$, such that $P$ occurs at positions $i$ and $j$ in $S$ and there is no occurrence of $P$ between $i$ and $j$, and their distance is defined as $j-i$. Given a pattern $P$ and a parameter $k$, the goal is to report the top-$k$ consecutive occurrences of $P$ in $S$ of minimal distance. The challenge is to compactly represent $S$ while supporting queries in time close to the length of $P$ and $k$. We give three time-space trade-offs for the problem. Let $n$ be the length of $S$, $m$ the length of $P$, and $ε\in(0,1]$. Our first result achieves $O(n\log n)$ space and optimal query time of $O(m+k)$. Our second and third results achieve linear space and query times either $O(m+k^{1+ε})$ or $O(m + k \log^{1+ε} n)$. Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees. |
| title | String Indexing for Top-$k$ Close Consecutive Occurrences |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2007.04128 |