The (unweighted) tree edit distance problem for $n$ node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in $O(n ^ 3)$ time [Demaine, Mozes, Rossman, and Weimann, ICALP 2007]. The same paper also showed that $O(n ^ 3)$ is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly sub-cubic time under the APSP conjecture [Bringmann, Gawrychowski, Mozes, and Weimann, SODA 2018]. In this paper, we break the cubic barrier by showing an $O(n ^ {2.9546})$ time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly sub-cubic time [Bringmann, Grandoni, Saha, and Vassilevska Williams, FOCS 2016].
翻译:用于 $n 节点树的树编辑距离问题( 未加权) 树编辑问题 要求计算两个有节点标签的根树之间的差异。 十年前的当前最佳算法以美元( + 3) 时间运行。 同一文件还显示, 美元( + 3) 是使用所谓的分解战略进行任何算法的最佳运行时间, 也就是几乎所有已知的这个问题的算法。 这些算法也将用于加权树编辑距离问题。 而在APSP 的轮廓下, 无法真正在次基时间解决这个问题 [ Bringmann, Gawrychowski, Mozes 和 Weimann, SODA 2018] 。 在本文中, 我们通过显示 $( n { 2. 95. 4}) 来打破立方屏障, 用于未加权树的距离问题的时间算法。 我们认为一个相当的最大化问题, 并且使用一个动态的编程计划, 包括许多特殊的次基 的基质矩阵, 通过一个固定的直径( 直径) 方案, 直径( 直方) 直方( 直方) 直方) 直方( 直方) 直方), 能够解成成成正 。