The tree edit distance problem is a natural generalization of the classic
string edit distance problem. Given two ordered, edge-labeled trees T1 and
T2, the edit distance between T1 and T2 is defined as the minimum
total cost of operations that transform T1 into T2. In one operation, we
can contract an edge, split a vertex into two or change the label of an edge.
For the weighted version of the problem, where the cost of each operation
depends on the type of the operation and the label on the edge involved,
O(n3) time algorithms are known for both rooted and unrooted
trees. The existence of a truly subcubic O(n3−ϵ) time
algorithm is unlikely, as it would imply a truly subcubic algorithm for the
APSP problem. However, recently Mao (FOCS'21) showed that if we assume that
each operation has a unit cost, then the tree edit distance between two rooted
trees can be computed in truly subcubic time. In this paper, we show how to
adapt Mao's algorithm to make it work for unrooted trees and we show an
O(n(7ω+15)/(2ω+6))≤O(n2.9417) time algorithm for the unweighted tree edit distance
between two unrooted trees, where ω≤2.373 is the matrix
multiplication exponent. It is the first known subcubic algorithm for unrooted
trees. The main idea behind our algorithm is the fact that to compute the tree
edit distance between two unrooted trees, it is enough to compute the tree edit
distance between an arbitrary rooting of the first tree and every rooting of
the second tree.Comment: 20 page