New and Improved Algorithms for Unordered Tree Inclusion

TL;DR

This paper introduces a new algorithm for the unordered tree inclusion problem that significantly improves the exponential factor in its runtime, enabling more efficient pattern matching in unordered trees.

Contribution

The authors develop a novel dynamic programming algorithm that reduces the exponential complexity from 2^{2d} to 2^d, advancing the computational efficiency for unordered tree inclusion.

Findings

New algorithm runs in O(2^d mn^2) time, improving previous methods.

The approach considers specific ancestor-descendant relationships for efficiency.

Restricted variants of the problem are also analyzed.

Abstract

The tree inclusion problem is, given two node-labeled trees $P$ and $T$ (the ``pattern tree'' and the ``target tree''), to locate every minimal subtree in $T$ (if any) that can be obtained by applying a sequence of node insertion operations to $P$. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is a classic algorithm by Kilpel\"{a}inen and Mannila from 1995 that runs in $O(2^{2d} mn)$ time, where $m$ and $n$ are the sizes of the pattern and target trees, respectively, and $d$ is the degree of the pattern tree. Here, we develop a new algorithm that runs in $O(2^{d} mn^2)$ time, improving the exponential factor from $2^{2d}$ to $2^d$ by considering a particular type of ancestor-descendant relationships that is suitable for dynamic programming. We also study restricted variants of the unordered tree inclusion problem.