Longest Common Extensions in Trees

Abstract

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.

• ${\mathrm {LCE}_{ PP }}(v_1, w_1, v_2, w_2)$: (path-path ${\mathrm {LCE}}$) return the longest common prefix of the paths $v_1 \leadsto w_1$ and $v_2 \leadsto w_2$.

• ${\mathrm {LCE}_{ PT }}(v_1, w_1, v_2)$: (path-tree ${\mathrm {LCE}}$) return maximal path-path LCE of the path $v_1 \leadsto w_1$ and any path from $v_2$ to a descendant leaf.

• ${\mathrm {LCE}_{ TT }}(v_1, v_2)$: (tree-tree ${\mathrm {LCE}}$) 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 ${\mathrm {LCE}_{ PP }}$ queries, we present a linear-space solution with $O(\log ^{*} n)$ query time. For ${\mathrm {LCE}_{ PT }}$ 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 $O(n \text {polylog}(n))$ must necessarily use ${\varOmega }(\log \log n)$ time to answer queries. For ${\mathrm {LCE}_{ TT }}$ queries, we present a time-space trade-off, that given any parameter $\tau$, $1 \le \tau \le n$, leads to an $O(n\tau )$ space and $O(n/\tau )$ 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.

P. Bille and I. L. Gørtz—Partially supported by the Danish Research Council (DFF – 4005-00267, DFF – 1323-00178) and the Advanced Technology Foundation.

P. Gawrychowski—Work done while the author held a post-doctoral position at the Warsaw Center of Mathematics and Computer Science.

G. M. Landau—Partially supported by the National Science Foundation Award 0904246, Israel Science Foundation grant 571/14, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.

O. Weimann—Partially supported by the Israel Science Foundation grant 794/13.