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

TL;DR

The paper addresses efficient construction of small-space string indexes, namely sparse suffix trees and LCE indexes, for readonly strings by leveraging a deterministic locally consistent parsing framework built on $\\tau$-partitioning sets. It combines a Cole–Vishkin style partitioning method with Je\\'z recompression to produce $\\tau$-partitioning sets of size $O(b)$ with $\\tau = n/b$, enabling an $O(b)$ space overhead on top of the input and near-linear construction time; for $b \\ge n^{\\varepsilon}$ this yields linear-time construction. The core results show that, for $\\tau$ in a broad range (including $\\tau \\ge 4$ up to $O(n/\\log^2 n)$), one can deterministically construct SSTs and LCE indexes in $\\mathcal{O}(n \\log_b n)$ time using $O(b)$ space, which subsumes and improves prior small-space deterministic constructions. The approach supports linear-time, linear-space ends for $b \\ge n^{\\varepsilon}$ and provides a general framework with tight time-space trade-offs, contributing a practical path toward optimal small-space string indexing under readonly inputs.

Abstract

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.

Figures (2)

• Figure 1: The $k$th phase. The heights of the dashed lines over $j_h$ are equal to $v_h$. Encircled positions are put into $S_k$: they are local minima of $v_h$, or are at the "boundaries" of all-$R$ regions, or form a gap of length ${>}2^k$. In the figure $R(j_{16}),\ldots, R(j_{20})$ hold and $R(j_{21})$ does not hold.
• Figure 2: The scheme generating $a_p$ via $\mathop{\mathsf{vbit}}$ reductions. If a node $\hat{t}$ has ingoing edges labeled with $\tilde{t}, \tilde{t}_1, \tilde{t}_2, \ldots, \tilde{t}_r$ (from left to right), then $\hat{t}$ encodes a tuple $\langle \tilde{w}_1, \tilde{w}_2, \ldots, \tilde{w}_\ell \rangle$ such that, for $j \in [1..r]$, $\tilde{w}_j = \mathop{\mathsf{vbit}}(\tilde{t}, \tilde{t}_j)$ and, for $j \in (r..\ell]$, $\tilde{w}_j = \infty$. In the figure, the numbers $t, t_1, t_2, \ldots, t_{m+5}$ correspond to consecutive positions $p, p_1, p_2, \ldots, p_{m+5}$ in the set $S'$, respectively. By looking at which of the ingoing edges are present and which are not, one can deduce that here we have $S' \cap (p..p{+}\tau/2^5] = \{p_1, \ldots, p_m\}$, $S' \cap (p_1..p_1{+}\tau/2^5] = \{p_2, \ldots, p_m\}$, $S' \cap (p_2..p_2{+}\tau/2^5] = \{p_3, \ldots, p_m, p_{m+1}\}$, $S' \cap (p_m..p_m{+}\tau/2^5] = \{p_{m+1}\}$, $S' \cap (p_{m+1}..p_{m+1}{+}\tau/2^5] = \{p_{m+2}, p_{m+3}\}$, $S' \cap (p_{m+2}..p_{m+2}{+}\tau/2^5] = \{p_{m+3}\}$, $S' \cap (p_{m+3}..p_{m+3}{+}\tau/2^5] = \{p_{m+4}, p_{m+5}\}$.

Theorems & Definitions (26)

• Lemma 1
• Theorem 2
• Lemma 2
• Theorem 3
• Theorem 4
• Lemma 5: see ColeVishkin
• Lemma 6: see ColeVishkin
• Lemma 6
• Lemma 6
• Lemma 6