An Algorithm for Comparing Similarity Between Two Trees

TL;DR

This paper introduces algorithms for computing the gapped edit distance between trees, enabling shape comparison in geometric computing, with efficient solutions for binary trees and methods applicable to 3D terrain contour trees.

Contribution

It presents a polynomial-time algorithm for gapped edit distance between binary trees and a new approach for complete subtree gap edit distance applicable to 3D contour trees.

Findings

Polynomial-time algorithm for binary trees' gap edit distance

NP-hardness of arbitrary tree comparison established

Algorithm for subtree gap edit distance in terrain analysis

Abstract

An important problem in geometric computing is defining and computing similarity between two geometric shapes, e.g. point sets, curves and surfaces, etc. Important geometric and topological information of many shapes can be captured by defining a tree structure on them (e.g. medial axis and contour trees). Hence, it is natural to study the problem of comparing similarity between trees. We study gapped edit distance between two ordered labeled trees, first proposed by Touzet \cite{Touzet2003}. Given two binary trees $T_{1}$ and $T_{2}$ with $m$ and $n$ nodes. We compute the general gap edit distance in $O(m^{3}n^{2} + m^{2}n^{3})$ time. The computation of this distance in the case of arbitrary trees has shown to be NP-hard \cite{Touzet2003}. We also give an algorithm for computing the complete subtree gap edit distance, which can be applied to comparing contour trees of terrains in $\mathbb{R}^{3}$.