Subtree Isomorphism Revisited

TL;DR

This paper explores the computational complexity of the Subtree Isomorphism problem, establishing conditional lower bounds based on SETH and presenting new algorithms for specific tree cases, advancing understanding of its algorithmic limits.

Contribution

It provides the first SETH-based lower bounds for various tree subclasses and introduces new algorithms for degree-bounded, shallow trees.

Findings

Truly subquadratic algorithms for binary trees of depth O(log log n) refute SETH.

Conditional lower bounds for binary and shallow rooted trees under SETH.

A new randomized algorithm with subquadratic complexity for degree-d trees of certain depths.

Abstract

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that: -- Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH. -- Even for rooted trees of depth $O(\log\log{n})$, where $n$ is the total number of vertices, a truly subquadratic algorithm refutes SETH. -- For every constant $d$, there is a constant $\epsilon_d>0$ and a randomized, truly subquadratic algorithm for degree-$d$ rooted trees of depth at most $(1+ \epsilon_d) \log_{d}{n}$. In particular, there is an $O(\min\{ 2.85^h ,n^2 \})$ algorithm for binary trees of depth $h$. Our reductions utilize new "tree gadgets" that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.