An algorithm for finding the largest approximately common substructures of two trees

Ordered, labeled trees are trees in which each node has a label and the left-to-right order of its children (if it has any) is fixed. Such trees have many applications in vision, pattern recognition, molecular biology and natural language processing. We consider a substructure of an ordered labeled tree T to be a connected subgraph of T. Given two ordered labeled trees T₁ and T₂ and an integer d, the largest approximately common substructure problem is to find a substructure U₁ of T₁ and a substructure U₂ of T₂ such that U₁ is within edit distance d of U₂ and where there does not exist any other substructure V₁ of T₁ and V₂ of T₂ such that V₁ and V₂ satisfy the distance constraint and the sum of the sizes of V₁ and V₂ is greater than the sum of the sizes of U₁ and U₂. We present a dynamic programming algorithm to solve this problem, which runs as fast as the fastest known algorithm for computing the edit distance of two trees when the distance allowed in the common substructures is a constant independent of the input trees. To demonstrate the utility of our algorithm, we discuss its application to discovering motifs in multiple RNA secondary structures (which are ordered labeled trees)