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 v 1 , v 2 , w 1 , and w 2 be nodes of T such that w 1 and w 2 are descendants of v 1 and v 2 respectively. LCE PP ( v 1 , w 1 , v 2 , w 2 ) : (path-path LCE) return the longest common prefix of the paths v 1 ź w 1 and v 2 ź w 2 . LCE PT ( v 1 , w 1 , v 2 ) : (path-tree LCE) return maximal path-path LCE of the path v 1 ź w 1 and any path from v 2 to a descendant leaf. LCE TT ( v 1 , v 2 ) : (tree-tree LCE) return a maximal path-path LCE of any pair of paths from v 1 and v 2 to descendant leaves. We present the first non-trivial bounds for supporting these queries. For LCE PP queries, we present a linear-space solution with O ( log * ź n ) query time. For LCE PT queries, we present a linear-space solution with O ( ( log ź log ź 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 ź ( log ź log ź n ) time to answer queries. For LCE TT queries, we present a time-space trade-off, that given any parameter ź, 1 ź ź ź n , leads to an O ( n ź ) space and O ( n / ź ) 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.