Nikita Sivukhin | Searcharxiv

arXiv2024

Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space

The notions of synchronizing and partitioning sets are recently introduced variants of locally consistent parsings with great potential in problem-solving. In this paper we propose a deterministic algorithm that constructs for a given readonly string of length $n$ over the alphabet $\{0,1,\ldots,n^{\mathcal{O}(1)}\}$ a variant of $τ$-partitioning set with size $\mathcal{O}(b)$ and $τ= \frac{n}{b}$ using $\mathcal{O}(b)$ space and $\mathcal{O}(\frac{1}εn)$ time provided $b \ge n^ε$, for $ε> 0$. As a corollary, for $b \ge n^ε$ and constant $ε> 0$, we obtain linear construction algorithms with $\mathcal{O}(b)$ space on top of the string for two major small-space indexes: a sparse suffix tree, which is a compacted trie built on $b$ chosen suffixes of the string, and a longest common extension (LCE) index, which occupies $\mathcal{O}(b)$ space and allows us to compute the longest common prefix for any pair of substrings in $\mathcal{O}(n/b)$ time. For both, the $\mathcal{O}(b)$ construction storage is asymptotically optimal since the tree itself takes $\mathcal{O}(b)$ space and any LCE index with $\mathcal{O}(n/b)$ query time must occupy at least $\mathcal{O}(b)$ space by a known trade-off (at least for $b \ge Ω(n / \log n)$). In case of arbitrary $b \ge Ω(\log^2 n)$, we present construction algorithms for the partitioning set, sparse suffix tree, and LCE index with $\mathcal{O}(n\log_b n)$ running time and $\mathcal{O}(b)$ space, thus also improving the state of the art.

cs.DS↗

arXiv2019

Compressed Multiple Pattern Matching

Given $d$ strings over the alphabet $\{0,1,\ldots,σ{-}1\}$, the classical Aho--Corasick data structure allows us to find all $occ$ occurrences of the strings in any text $T$ in $O(|T| + occ)$ time using $O(m\log m)$ bits of space, where $m$ is the number of edges in the trie containing the strings. Fix any constant $\varepsilon \in (0, 2)$. We describe a compressed solution for the problem that, provided $σ\le m^δ$ for a constant $δ< 1$, works in $O(|T| \frac{1}{\varepsilon} \log\frac{1}{\varepsilon} + occ)$ time, which is $O(|T| + occ)$ since $\varepsilon$ is constant, and occupies $mH_k + 1.443 m + \varepsilon m + O(d\log\frac{m}{d})$ bits of space, for all $0 \le k \le \max\{0,α\log_σm - 2\}$ simultaneously, where $α\in (0,1)$ is an arbitrary constant and $H_k$ is the $k$th-order empirical entropy of the trie. Hence, we reduce the $3.443m$ term in the space bounds of previously best succinct solutions to $(1.443 + \varepsilon)m$, thus solving an open problem posed by Belazzougui. Further, we notice that $L = \log\binom{σ(m+1)}{m} - O(\log(σm))$ is a worst-case space lower bound for any solution of the problem and, for $d = o(m)$ and constant $\varepsilon$, our approach allows to achieve $L + \varepsilon m$ bits of space, which gives an evidence that, for $d = o(m)$, the space of our data structure is theoretically optimal up to the $\varepsilon m$ additive term and it is hardly possible to eliminate the term $1.443m$. In addition, we refine the space analysis of previous works by proposing a more appropriate definition for $H_k$. We also simplify the construction for practice adapting the fixed block compression boosting technique, then implement our data structure, and conduct a number of experiments showing that it is comparable to the state of the art in terms of time and is superior in space.

cs.DS↗