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Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space (2105.03782v3)

Published 8 May 2021 in cs.DS

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 $\tau$-partitioning set with size $\mathcal{O}(b)$ and $\tau = \frac{n}{b}$ using $\mathcal{O}(b)$ space and $\mathcal{O}(\frac{1}{\epsilon}n)$ time provided $b \ge n\epsilon$, for $\epsilon > 0$. As a corollary, for $b \ge n\epsilon$ and constant $\epsilon > 0$, 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 $b \ge \Omega(n / \log n)$). In case of arbitrary $b \ge \Omega(\log2 n)$, 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.

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