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 v1v1, v2v2, w1w1, and w2w2 be nodes of T   such that w1w1 and w2w2 are descendants of v1v1 and v2v2 respectively.
• LCEPP(v1,w1,v2,w2)LCEPP(v1,w1,v2,w2): (path–path LCE) return the longest common prefix of the paths v1↝w1v1↝w1 and v2↝w2v2↝w2.
• LCEPT(v1,w1,v2)LCEPT(v1,w1,v2): (path–tree LCE) return maximal path–path LCE of the path v1↝w1v1↝w1 and any path from v2v2 to a descendant leaf.
• LCETT(v1,v2)LCETT(v1,v2): (tree–tree LCE) return a maximal path–path LCE of any pair of paths from v1v1 and v2v2 to descendant leaves. We present the first non-trivial bounds for supporting these queries. For LCEPPLCEPP queries, we present a linear-space solution with O(log⁎⁡n)O(log⁎⁡n) query time. For LCEPTLCEPT queries, we present a linear-space solution with O((log⁡log⁡n)2)O((log⁡log⁡n)2) query time, and complement this with a lower bound showing that any path–tree LCE structure of size O(npolylog(n))O(npolylog(n)) must necessarily use Ω(log⁡log⁡n)Ω(log⁡log⁡n) time to answer queries. For LCETTLCETT queries, we present a time-space trade-off, that given any parameter τ  , 1≤τ≤n1≤τ≤n, leads to an O(nτ)O(nτ) space and O(n/τ)O(n/τ) query-time solution (all of these bounds hold on a standard unit-cost RAM model with logarithmic word size). This is complemented with a reduction from the set intersection problem implying that a fast linear space solution is not likely to exist.