Longest common extensions in trees

Philip Bille, Paweł Gawrychowski, Inge Li Gørtz, Gad M. Landau, Oren Weimann

Research output: Contribution to journalArticlepeer-review


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 LCEPT 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(npolylog(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 (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.

Original languageEnglish
Pages (from-to)98-107
Number of pages10
JournalTheoretical Computer Science
StatePublished - 25 Jul 2016

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V.


  • Longest common prefix
  • Pattern matching in trees
  • Suffix tree of a tree

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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