一种树相似度比较算法
An Algorithm for Comparing Similarity Between Two Trees
摘要：
几何计算中的一个重要问题是定义和计算两个几何形状之间的相似性，例如点集、曲线和曲面等。许多形状的重要几何和拓扑信息可以通过在它们上定义树结构（例如中轴树和轮廓树）来捕获。因此，研究树之间的相似性比较问题是很自然的。我们研究了Touzet \cite{Touzet2003}首先提出的两棵有序标记树之间的间隙编辑距离。给出了两棵二叉树$T_{1}$和$T_{2}$，其中节点为$m$和$n$。我们在$O(m^{3}n^{2} + m^{2}n^{3})$时间上计算一般间隙编辑距离。在任意树的情况下，该距离的计算已被证明是NP难的\cite{Touzet2003}。本文还给出了一个计算完全子树间隙编辑距离的算法，该算法可用于$\mathbb{R}^{3}$中地形等高线树的比较。
英文摘要：
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 $\mathbb{R}^{3}$.
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