Tight bounds for the sensitivity of CDAWGs with left-end edits
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915121002971136 |
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| author | Fujimaru, Hiroto Nakashima, Yuto Inenaga, Shunsuke |
| author_facet | Fujimaru, Hiroto Nakashima, Yuto Inenaga, Shunsuke |
| contents | Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string $T$, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if $e$ is the number of edges of the CDAWG for string $T$, then the number of new edges added to the CDAWG after a left-end edit operation on $T$ does not exceed $e$. Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires $Ω(n^2)$ time for some string of length $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_01726 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tight bounds for the sensitivity of CDAWGs with left-end edits Fujimaru, Hiroto Nakashima, Yuto Inenaga, Shunsuke Data Structures and Algorithms Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string $T$, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if $e$ is the number of edges of the CDAWG for string $T$, then the number of new edges added to the CDAWG after a left-end edit operation on $T$ does not exceed $e$. Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires $Ω(n^2)$ time for some string of length $n$. |
| title | Tight bounds for the sensitivity of CDAWGs with left-end edits |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2303.01726 |