TY - GEN

T1 - Tree edit distance cannot be computed in strongly subcubic time (unless APSP can)

AU - Bringmann, Karl

AU - Gawrychowski, Pawel

AU - Mozes, Shay

AU - Weimann, Oren

N1 - Publisher Copyright: © Copyright 2018 by SIAM.

PY - 2018

Y1 - 2018

N2 - The edit distance between two rooted ordered trees with n nodes labeled from an alphabet Σ is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. Tree edit distance is a well known generalization of string edit distance. The fastest known algorithm for tree edit distance runs in cubic O(n3) time and is based on a similar dynamic programming solution as string edit distance. In this paper we show that a truly subcubic O(n3-ϵ) time algorithm for tree edit distance is unlikely: For Σ = (n), a truly subcubic algorithm for tree edit distance implies a truly subcubic algorithm for the all pairs shortest paths problem. For Σ = O(1), a truly subcubic algorithm for tree edit distance implies an O(nk-ϵ) algorithm for finding a maximum weight k-clique. Thus, while in terms of upper bounds string edit distance and tree edit distance are highly related, in terms of lower bounds string edit distance exhibits the hardness of the strong exponential time hypothesis [Backurs, Indyk STOC'15] whereas tree edit distance exhibits the hardness of all pairs shortest paths. Our result provides a matching conditional lower bound for one of the last remaining classic dynamic programming problems.

AB - The edit distance between two rooted ordered trees with n nodes labeled from an alphabet Σ is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. Tree edit distance is a well known generalization of string edit distance. The fastest known algorithm for tree edit distance runs in cubic O(n3) time and is based on a similar dynamic programming solution as string edit distance. In this paper we show that a truly subcubic O(n3-ϵ) time algorithm for tree edit distance is unlikely: For Σ = (n), a truly subcubic algorithm for tree edit distance implies a truly subcubic algorithm for the all pairs shortest paths problem. For Σ = O(1), a truly subcubic algorithm for tree edit distance implies an O(nk-ϵ) algorithm for finding a maximum weight k-clique. Thus, while in terms of upper bounds string edit distance and tree edit distance are highly related, in terms of lower bounds string edit distance exhibits the hardness of the strong exponential time hypothesis [Backurs, Indyk STOC'15] whereas tree edit distance exhibits the hardness of all pairs shortest paths. Our result provides a matching conditional lower bound for one of the last remaining classic dynamic programming problems.

UR - http://www.scopus.com/inward/record.url?scp=85038622175&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/1.9781611975031.77

DO - https://doi.org/10.1137/1.9781611975031.77

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1190

EP - 1206

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -