arXiv subjects

Inge Li Gørtz

Publications and source records attributed to Inge Li Gørtz.

48 records · Page 3Linked to original sources

arXiv2017

Practical and Effective Re-Pair Compression

Re-Pair is an efficient grammar compressor that operates by recursively replacing high-frequency character pairs with new grammar symbols. The most space-efficient linear-time algorithm computing Re-Pair uses $(1+ε)n+\sqrt n$ words on top of the re-writable text (of length $n$ and stored in $n$ words), for any constant $ε>0$; in practice however, this solution uses complex sub-procedures preventing it from being practical. In this paper, we present an implementation of the above-mentioned result making use of more practical solutions; our tool further improves the working space to $(1.5+ε)n$ words (text included), for some small constant $ε$. As a second contribution, we focus on compact representations of the output grammar. The lower bound for storing a grammar with $d$ rules is $\log(d!)+2d\approx d\log d+0.557 d$ bits, and the most efficient encoding algorithm in the literature uses at most $d\log d + 2d$ bits and runs in $\mathcal O(d^{1.5})$ time. We describe a linear-time heuristic maximizing the compressibility of the output Re-Pair grammar. On real datasets, our grammar encoding uses---on average---only $2.8\%$ more bits than the information-theoretic minimum. In half of the tested cases, our compressor improves the output size of 7-Zip with maximum compression rate turned on.

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arXiv2016

Deterministic Indexing for Packed Strings

Given a string $S$ of length $n$, the classic string indexing problem is to preprocess $S$ into a compact data structure that supports efficient subsequent pattern queries. In the \emph{deterministic} variant the goal is to solve the string indexing problem without any randomization (at preprocessing time or query time). In the \emph{packed} variant the strings are stored with several character in a single word, giving us the opportunity to read multiple characters simultaneously. Our main result is a new string index in the deterministic \emph{and} packed setting. Given a packed string $S$ of length $n$ over an alphabet $σ$, we show how to preprocess $S$ in $O(n)$ (deterministic) time and space $O(n)$ such that given a packed pattern string of length $m$ we can support queries in (deterministic) time $O\left(m/α+ \log m + \log \log σ\right), $ where $α= w / \log σ$ is the number of characters packed in a word of size $w = Θ(\log n)$. Our query time is always at least as good as the previous best known bounds and whenever several characters are packed in a word, i.e., $\log σ\ll w$, the query times are faster.

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arXiv2016

Finger Search in Grammar-Compressed Strings

Grammar-based compression, where one replaces a long string by a small context-free grammar that generates the string, is a simple and powerful paradigm that captures many popular compression schemes. Given a grammar, the random access problem is to compactly represent the grammar while supporting random access, that is, given a position in the original uncompressed string report the character at that position. In this paper we study the random access problem with the finger search property, that is, the time for a random access query should depend on the distance between a specified index $f$, called the \emph{finger}, and the query index $i$. We consider both a static variant, where we first place a finger and subsequently access indices near the finger efficiently, and a dynamic variant where also moving the finger such that the time depends on the distance moved is supported. Let $n$ be the size the grammar, and let $N$ be the size of the string. For the static variant we give a linear space representation that supports placing the finger in $O(\log N)$ time and subsequently accessing in $O(\log D)$ time, where $D$ is the distance between the finger and the accessed index. For the dynamic variant we give a linear space representation that supports placing the finger in $O(\log N)$ time and accessing and moving the finger in $O(\log D + \log \log N)$ time. Compared to the best linear space solution to random access, we improve a $O(\log N)$ query bound to $O(\log D)$ for the static variant and to $O(\log D + \log \log N)$ for the dynamic variant, while maintaining linear space. As an application of our results we obtain an improved solution to the longest common extension problem in grammar compressed strings. To obtain our results, we introduce several new techniques of independent interest, including a novel van Emde Boas style decomposition of grammars.

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arXiv2016

Space-Efficient Re-Pair Compression

Re-Pair is an effective grammar-based compression scheme achieving strong compression rates in practice. Let $n$, $σ$, and $d$ be the text length, alphabet size, and dictionary size of the final grammar, respectively. In their original paper, the authors show how to compute the Re-Pair grammar in expected linear time and $5n + 4σ^2 + 4d + \sqrt{n}$ words of working space on top of the text. In this work, we propose two algorithms improving on the space of their original solution. Our model assumes a memory word of $\lceil\log_2 n\rceil$ bits and a re-writable input text composed by $n$ such words. Our first algorithm runs in expected $\mathcal O(n/ε)$ time and uses $(1+ε)n +\sqrt n$ words of space on top of the text for any parameter $0<ε\leq 1$ chosen in advance. Our second algorithm runs in expected $\mathcal O(n\log n)$ time and improves the space to $n +\sqrt n$ words.

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arXiv2016

Dynamic Relative Compression, Dynamic Partial Sums, and Substring Concatenation

Given a static reference string $R$ and a source string $S$, a relative compression of $S$ with respect to $R$ is an encoding of $S$ as a sequence of references to substrings of $R$. Relative compression schemes are a classic model of compression and have recently proved very successful for compressing highly-repetitive massive data sets such as genomes and web-data. We initiate the study of relative compression in a dynamic setting where the compressed source string $S$ is subject to edit operations. The goal is to maintain the compressed representation compactly, while supporting edits and allowing efficient random access to the (uncompressed) source string. We present new data structures that achieve optimal time for updates and queries while using space linear in the size of the optimal relative compression, for nearly all combinations of parameters. We also present solutions for restricted and extended sets of updates. To achieve these results, we revisit the dynamic partial sums problem and the substring concatenation problem. We present new optimal or near optimal bounds for these problems. Plugging in our new results we also immediately obtain new bounds for the string indexing for patterns with wildcards problem and the dynamic text and static pattern matching problem.

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arXiv2016

Subsequence Automata with Default Transitions

Let $S$ be a string of length $n$ with characters from an alphabet of size $σ$. The \emph{subsequence automaton} of $S$ (often called the \emph{directed acyclic subsequence graph}) is the minimal deterministic finite automaton accepting all subsequences of $S$. A straightforward construction shows that the size (number of states and transitions) of the subsequence automaton is $O(nσ)$ and that this bound is asymptotically optimal. In this paper, we consider subsequence automata with \emph{default transitions}, that is, special transitions to be taken only if none of the regular transitions match the current character, and which do not consume the current character. We show that with default transitions, much smaller subsequence automata are possible, and provide a full trade-off between the size of the automaton and the \emph{delay}, i.e., the maximum number of consecutive default transitions followed before consuming a character. Specifically, given any integer parameter $k$, $1 < k \leq σ$, we present a subsequence automaton with default transitions of size $O(nk\log_{k}σ)$ and delay $O(\log_k σ)$. Hence, with $k = 2$ we obtain an automaton of size $O(n \log σ)$ and delay $O(\log σ)$. On the other extreme, with $k = σ$, we obtain an automaton of size $O(n σ)$ and delay $O(1)$, thus matching the bound for the standard subsequence automaton construction. Finally, we generalize the result to multiple strings. The key component of our result is a novel hierarchical automata construction of independent interest.

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arXiv2015

Distance labeling schemes for trees

We consider distance labeling schemes for trees: given a tree with $n$ nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg (J. Graph Theory 2000) establish that labels must use $Θ(\log^2 n)$ bits\footnote{Throughout this paper we use $\log$ for $\log_2$.}. Gavoille et. al. (ESA 2001) show that for very small approximate stretch, labels use $Θ(\log n \log \log n)$ bits. Several other papers investigate various variants such as, for example, small distances in trees (Alstrup et. al., SODA'03). We improve the known upper and lower bounds of exact distance labeling by showing that $\frac{1}{4} \log^2 n$ bits are needed and that $\frac{1}{2} \log^2 n$ bits are sufficient. We also give ($1+ε$)-stretch labeling schemes using $Θ(\log n)$ bits for constant $ε>0$. ($1+ε$)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar (STOC 2004). In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size $2\log n - Θ(\log\log n)$. For simple paths with $k$ nodes and edge weights in $[1,n]$, we show that labels must have size $\frac{k-1}{k}\log n+Θ(\log k)$.

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arXiv2015

Longest Common Extensions in Sublinear Space

The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports LCE$(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $O(n)$ space and $O(1)$ query time. In this paper we show that for any trade-off parameter $1 \leq τ\leq n$, the problem can be solved in $O(\frac{n}τ)$ space and $O(τ)$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.

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arXiv2014

Compact q-gram Profiling of Compressed Strings

We consider the problem of computing the q-gram profile of a string \str of size $N$ compressed by a context-free grammar with $n$ production rules. We present an algorithm that runs in $O(N-α)$ expected time and uses $O(n+q+\kq)$ space, where $N-α\leq qn$ is the exact number of characters decompressed by the algorithm and $\kq\leq N-α$ is the number of distinct q-grams in $\str$. This simultaneously matches the current best known time bound and improves the best known space bound. Our space bound is asymptotically optimal in the sense that any algorithm storing the grammar and the q-gram profile must use $Ω(n+q+\kq)$ space. To achieve this we introduce the q-gram graph that space-efficiently captures the structure of a string with respect to its q-grams, and show how to construct it from a grammar.

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arXiv2014

Compressed Subsequence Matching and Packed Tree Coloring

We present a new algorithm for subsequence matching in grammar compressed strings. Given a grammar of size $n$ compressing a string of size $N$ and a pattern string of size $m$ over an alphabet of size $σ$, our algorithm uses $O(n+\frac{nσ}{w})$ space and $O(n+\frac{nσ}{w}+m\log N\log w\cdot occ)$ or $O(n+\frac{nσ}{w}\log w+m\log N\cdot occ)$ time. Here $w$ is the word size and $occ$ is the number of occurrences of the pattern. Our algorithm uses less space than previous algorithms and is also faster for $occ=o(\frac{n}{\log N})$ occurrences. The algorithm uses a new data structure that allows us to efficiently find the next occurrence of a given character after a given position in a compressed string. This data structure in turn is based on a new data structure for the tree color problem, where the node colors are packed in bit strings.

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arXiv2013

Fingerprints in Compressed Strings

The Karp-Rabin fingerprint of a string is a type of hash value that due to its strong properties has been used in many string algorithms. In this paper we show how to construct a data structure for a string $S$ of size $N$ compressed by a context-free grammar of size $n$ that answers fingerprint queries. That is, given indices $i$ and $j$, the answer to a query is the fingerprint of the substring $S[i,j]$. We present the first O(n) space data structures that answer fingerprint queries without decompressing any characters. For Straight Line Programs (SLP) we get $O(\log N)$ query time, and for Linear SLPs (an SLP derivative that captures LZ78 compression and its variations) we get $O(\log \log N)$ query time. Hence, our data structures has the same time and space complexity as for random access in SLPs. We utilize the fingerprint data structures to solve the longest common extension problem in query time $O(\log N \log \lce)$ and $O(\log \lce \log\log \lce + \log\log N)$ for SLPs and Linear SLPs, respectively. Here, $\lce$ denotes the length of the LCE.

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arXiv2012

Sparse Suffix Tree Construction with Small Space

We consider the problem of constructing a sparse suffix tree (or suffix array) for $b$ suffixes of a given text $T$ of size $n$, using only $O(b)$ words of space during construction time. Breaking the naive bound of $Ω(nb)$ time for this problem has occupied many algorithmic researchers since a different structure, the (evenly spaced) sparse suffix tree, was introduced by K{ä}rkk{ä}inen and Ukkonen in 1996. While in the evenly spaced sparse suffix tree the suffixes considered must be evenly spaced in $T$, here there is no constraint on the locations of the suffixes. We show that the sparse suffix tree can be constructed in $O(n\log^2b)$ time. To achieve this we develop a technique, which may be of independent interest, that allows to efficiently answer $b$ longest common prefix queries on suffixes of $T$, using only $O(b)$ space. We expect that this technique will prove useful in many other applications in which space usage is a concern. Furthermore, additional tradeoffs between the space usage and the construction time are given.

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