The edit distance between two ordered trees with vertex labels 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. In this paper, we present a worstcase O(n 3 )-time algorithm for this problem, improving the previous best O(n 3 log n)time algorithm [6]. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems (which is interesting in its own right), together with a deeper understanding of the previous algorithms for the problem. We also prove the optimality of our algorithm among the family of decomposition strategy algorithms-which also includes the previous fastest algorithms-by tightening the known lower bound of Ω(n 2 log 2 n) [4] to Ω(n 3 ), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds of Θ(nm 2 (1 + log n m )) when the two trees have different sizes m and n, where m < n.

The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the Loś-Tarski theorem and Lyndon's positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existentialpositive formula of equal quantifier-rank.

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The edit distance between two ordered rooted trees with vertex labels 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. In this article, we present a worst-case O(n 3 )-time algorithm for the problem when the two trees have size n, improving the previous best O(n 3 log n)-time algorithm. Our result requires a novel adaptive strategy for deciding how a dynamic program divides into subproblems, together with a deeper understanding of the previous algorithms for the problem. We prove the optimality of our algorithm among the family of decomposition strategy algorithms-which also includes the previous fastest algorithms-by tightening the known lower bound of (n 2 log 2 n) to (n 3 ), matching our algorithm's running time. Furthermore, we obtain matching upper and lower bounds for decomposition strategy algorithms of (nm 2 (1 + log n m )) when the two trees have sizes m and n and m < n.

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