Tree edit distance is a well-studied measure of dissimilarity between rooted trees with node labels. It can be computed in $O(n^3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007], and fine-grained hardness results suggest that the weighted version of this problem cannot be solved in truly subcubic time unless the APSP conjecture is false [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. We consider the unweighted version of tree edit distance, where every insertion, deletion, or relabeling operation has unit cost. Given a parameter $k$ as an upper bound on the distance, the previous fastest algorithm for this problem runs in $O(nk^3)$ time [Touzet, CPM 2005], which improves upon the cubic-time algorithm for $k\ll n^{2/3}$. In this paper, we give a faster algorithm taking $O(nk^2 \log n)$ time, improving both of the previous results for almost the full range of $\log n \ll k\ll n/\sqrt{\log n}$.
翻译:树的编辑距离是具有节点标签的树根树之间不同之处的一种研究良好的衡量尺度。 可以用$O (n) 3 美元的时间计算 [ maine, Mozes, Rossman, and Weimann, CROCP 2007] 和细微的硬度结果显示, 这个问题的加权版本无法在真正的子立方时间解决, 除非 APSP 的猜测是假的 [ Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018] 。 我们考虑的是未加权的树编辑距离版本, 在那里, 每一次插入、 删除或重新标签操作都有单位成本。 鉴于参数是美元作为距离的上限, 这一问题之前最快的算法是 $(nk) 3 美元 [Touzet, CPM 2005], 时间在立方时算法得到改进 $k\ll n3} $。 在本文中, 我们给出了一种快速的算法, $(n_2\log n$) 时间, 改进了前两次结果的全程。