An Unambiguous And Complete Dynamic Programming Algorithm For Tree Alignment

Pairwise ordered tree alignment is an important pattern matching problem, motivated by RNA secondary structure comparison for example, that can be solved efficiently using Dynamic Programming (DP). An inquiry into the multiplicity of optimal solutions, as well as the existence of potentially interesting sub-optimal solutions, naturally motivates the question of exploring the space of all, optimal and sub-optimal, tree alignments. There are well-known DP-based techniques that allow for such an exploration, but they require completeness and unambiguity of the DP scheme, i.e. the existence of a score-preserving bijection between the search space and the set of possible derivations of the DP scheme. In this paper, we present the first unambiguous and complete dynamic programming algorithm for the alignment of a pair of ordered rooted trees. Our algorithm optimally aligns two trees of size n1 and n2 in Θ(n1 n2 max(n1, n2) ) time in the worst-case scenario. Assuming uniformly-drawn random trees as input, it has average-case time and space complexities in Θ(n1 n2).