On the hardness of computing the edit distance of shallow trees

Charalampopoulos, Panagiotis and Gawrychowski, P. and Mozes, S. and Weimann, O. (2022) On the hardness of computing the edit distance of shallow trees. Lecture Notes in Computer Science 13617 , pp. 290-302. ISSN 0302-9743.

Abstract

We consider the edit distance problem on rooted ordered trees parameterized by the trees’ depth. For two trees of size at most n and depth at most d, the state-of-the-art solutions of Zhang and Shasha [SICOMP 1989] and Demaine et al. [TALG 2009] have runtimes O(n^2 · d^2) and O(n^3), respectively, and are based on so-called decomposition algorithms. It has been recently shown by Bringmann et al. [TALG 2020] that, when d=Θ(n), one cannot compute the edit distance of two trees in O(n^{3−ϵ}) time (for any constant ϵ>0) under the APSP hypothesis. However, for small values of d, it is not known whether the O(n^2 · d^2) upper bound of Zhang and Shasha is optimal. We make the following twofold contribution. First, we show that under the APSP hypothesis there is no algorithm with runtime O(n^2 · d^{1−ϵ}) (for any constant ϵ>0) when d=poly(n). Second, we show that there is no decomposition algorithm that runs in time o(min{n^2 · d^2,n^3}).

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