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 $\tau$-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 $b \ge \Omega(\log^2 n)$, 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.