Construction of Sparse Suffix Trees and LCE Indexes in Optimal Time and Space
arxiv(2021)
摘要
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,…,n^𝒪(1)} a variant of τ-partitioning set with
size 𝒪(b) and τ = n/b using 𝒪(b) space and
𝒪(1/ϵn) time provided b ≥ n^ϵ, for
ϵ > 0. As a corollary, for b ≥ n^ϵ and constant ϵ >
0, we obtain linear construction algorithms with 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 𝒪(b) space and allows
us to compute the longest common prefix for any pair of substrings in
𝒪(n/b) time. For both, the 𝒪(b) construction storage is
asymptotically optimal since the tree itself takes 𝒪(b) space and
any LCE index with 𝒪(n/b) query time must occupy at least
𝒪(b) space by a known trade-off (at least for b ≥Ω(n /
log n)). In case of arbitrary b ≥Ω(log^2 n), we present
construction algorithms for the partitioning set, sparse suffix tree, and LCE
index with 𝒪(nlog_b n) running time and 𝒪(b) space,
thus also improving the state of the art.
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