Longest common extensions in trees

The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices. The LCE problem asks to preprocess a string into a compact data structure that supports fast LCE queries. In this paper we generalize the LCE problem to trees and suggest a few applications of LCE in trees to tries and XML databases. Given a labeled and rooted tree T of size n, the goal is to preprocess T into a compact data structure that support the following LCE queries between subpaths and subtrees in T. Let v₁, v₂, w₁, and w₂ be nodes of T such that w₁ and w₂ are descendants of v₁ and v₂ respectively. - LCEPP(v₁, w₁, v₂, w₂): (path-path LCE) return the longest common prefix of the paths v₁ ⇝ w₁ and v₂ ⇝ w₂. - LCEPT (v₁, w₁, v₂): (path-tree LCE) return maximal path-path LCE of the path v₁ ⇝ w₁ and any path from v₂ to a descendant leaf. - LCETT (v₁, v₂): (tree-tree LCE) return a maximal path-path LCE of any pair of paths from v₁ and v₂ to descendant leaves. We present the first non-trivial bounds for supporting these queries. For LCEPP queries, we present a linear-space solution with O(log^∗ n) query time. For LCE_PT queries, we present a linear-space solution with O((log log n)²) query time, and complement this with a lower bound showing that any path-tree LCE structure of size O(n polylog(n)) must necessarily use Ω(log log n) time to answer queries. For LCETT queries, we present a time-space trade-off, that given any parameter τ, 1 ≤ τ ≤ n, leads to an O(nτ) space and O(n/τ) query-time solution. This is complemented with a reduction to the set intersection problem implying that a fast linear space solution is not likely to exist.