arXiv subjects

Tooru Akagi

Publications and source records attributed to Tooru Akagi.

5 recordsLinked to original sources

arXiv2022

Minimal Absent Words on Run-Length Encoded Strings

A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur (as a substring) in $T$ and any proper substring of $w$ occurs in $T$. State-of-the-art data structures for reporting the set $\mathsf{MAW}(T)$ of MAWs from a given string $T$ of length $n$ require $O(n)$ space, can be built in $O(n)$ time, and can report all MAWs in $O(|\mathsf{MAW}(T)|)$ time upon a query. This paper initiates the problem of computing MAWs from a compressed representation of a string. In particular, we focus on the most basic compressed representation of a string, run-length encoding (RLE), which represents each maximal run of the same characters $a$ by $a^p$ where $p$ is the length of the run. Let $m$ be the RLE-size of string $T$. After categorizing the MAWs into five disjoint sets $\mathcal{M}_1$, $\mathcal{M}_2$, $\mathcal{M}_3$, $\mathcal{M}_4$, $\mathcal{M}_5$ using RLE, we present matching upper and lower bounds for the number of MAWs in $\mathcal{M}_i$ for $i = 1,2,4,5$ in terms of RLE-size $m$, except for $\mathcal{M}_3$ whose size is unbounded by $m$. We then present a compact $O(m)$-space data structure that can report all MAWs in optimal $O(|\mathsf{MAW}(T)|)$ time.

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arXiv2021

Sensitivity of string compressors and repetitiveness measures

The sensitivity of a string compression algorithm $C$ asks how much the output size $C(T)$ for an input string $T$ can increase when a single character edit operation is performed on $T$. This notion enables one to measure the robustness of compression algorithms in terms of errors and/or dynamic changes occurring in the input string. In this paper, we analyze the worst-case multiplicative sensitivity of string compression algorithms, which is defined by $\max_{T \in \Sigma^n}\{C(T')/C(T) : ed(T, T') = 1\}$, where $ed(T, T')$ denotes the edit distance between $T$ and $T'$. For the most common versions of the Lempel-Ziv 77 compressors, we prove that the worst-case multiplicative sensitivity is upper bounded by a small constant, and give matching lower bounds. We generalize these results to the smallest bidirectional scheme $b$. In addition, we show that the sensitivity of a grammar-based compressor called GCIS is also a small constant. Further, we extend the notion of the worst-case sensitivity to string repetitiveness measures such as the smallest string attractor size $\gamma$ and the substring complexity $\delta$, and show that the worst-case sensitivity of $\delta$ is also a small constant. These results contrast with the previously known related results such that the size $z_{\rm 78}$ of the Lempel-Ziv 78 factorization can increase by a factor of $\Omega(n^{1/4})$ [Lagarde and Perifel, 2018], and the number $r$ of runs in the Burrows-Wheeler transform can increase by a factor of $\Omega(\log n)$ [Giuliani et al., 2021] when a character is prepended to an input string of length $n$. By applying our sensitivity bounds of $\delta$ or the smallest grammar to known results (c.f. [Navarro, 2021]), some non-trivial upper bounds for the sensitivities of important string compressors and repetitiveness measures including $\gamma$, $r$, LZ-End, RePair, LongestMatch, and AVL-grammar are derived.

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arXiv2021

Grammar Index By Induced Suffix Sorting

Pattern matching is the most central task for text indices. Most recent indices leverage compression techniques to make pattern matching feasible for massive but highly-compressible datasets. Within this kind of indices, we propose a new compressed text index built upon a grammar compression based on induced suffix sorting [Nunes et al., DCC'18]. We show that this grammar exhibits a locality sensitive parsing property, which allows us to specify, given a pattern $P$, certain substrings of $P$, called cores, that are similarly parsed in the text grammar whenever these occurrences are extensible to occurrences of $P$. Supported by the cores, given a pattern of length $m$, we can locate all its $occ$ occurrences in a text $T$ of length $n$ within $O(m \lg |\mathcal{S}| + occ_C \lg|\mathcal{S}| \lg n + occ)$ time, where $\mathcal{S}$ is the set of all characters and non-terminals, $occ$ is the number of occurrences, and $occ_C$ is the number of occurrences of a chosen core $C$ of $P$ in the right hand side of all production rules of the grammar of $T$. Our grammar index requires $O(g)$ words of space and can be built in $O(n)$ time using $O(g)$ working space, where $g$ is the sum of the right hand sides of all production rules. We underline the strength of our grammar index with an exhaustive practical evaluation that gives evidence that our proposed solution excels at locating long patterns in highly-repetitive texts.

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arXiv2021

Combinatorics of minimal absent words for a sliding window

A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur in $T$ but the proper substrings of $w$ occur in $T$. For example, let $\Sigma = \{\mathtt{a, b, c}\}$ be the alphabet. Then, the set of MAWs for string $w = \mathtt{abaab}$ is $\{\mathtt{aaa, aaba, bab, bb, c}\}$. In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length $d$ is shifted over the input string $T$ of length $n$, where $1 \leq d < n$. We present \emph{tight} upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over $T$, both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].

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arXiv2019

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

A substring $u$ of a string $T$ is called a minimal unique substring (MUS) of $T$ if $u$ occurs exactly once in $T$ and any proper substring of $u$ occurs at least twice in $T$. A string $w$ is called a minimal absent word (MAW) of $T$ if $w$ does not occur in $T$ and any proper substring of $w$ occurs in $T$. In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string $T$. We first show how the set of MUSs can change in a sliding window over $T$, and present an $O(n\log\sigma)$-time and $O(d)$-space algorithm to compute MUSs in a sliding window of width $d$ over $T$, where $\sigma$ is the maximum number of distinct characters in every window. We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over $T$. Our bounds improve on the previous results in [Crochemore et al., 2017].

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