Unrooted unordered homeomorphic subtree alignment of RNA trees

We generalize some current approaches for RNA tree alignment, which are traditionally confined to ordered rooted mappings, to also consider unordered unrooted mappings. We define the Homeomorphic Subtree Alignment problem (HSA), and present a new algorithm which applies to several modes, combining global or local, ordered or unordered, and rooted or unrooted tree alignments. Our algorithm generalizes previous algorithms that either solved the problem in an asymmetric manner, or were restricted to the rooted and/or ordered cases. Focusing here on the most general unrooted unordered case, we show that for input trees T and S, our algorithm has an O(n(T)n(S) + min(d(T),d(S))L(T)L(S)) time complexity, where n(T),L(T) and d(T) are the number of nodes, the number of leaves, and the maximum node degree in T, respectively (satisfying d(T) ≤ L(T) ≤ n(T)), and similarly for n(S),L(S) and d(S) with respect to the tree S. This improves the time complexity of previous algorithms for less general variants of the problem. In order to obtain this time bound for HSA, we developed new algorithms for a generalized variant of the Min-Cost Bipartite Matching problem (MCM), as well as to two derivatives of this problem, entitled All-Cavity-MCM and All-Pairs-Cavity-MCM. For two input sets of size n and m, where n ≤ m, MCM and both its cavity derivatives are solved in O(n(3) + nm) time, without the usage of priority queues (e.g. Fibonacci heaps) or other complex data structures. This gives the first cubic time algorithm for All-Pairs-Cavity-MCM, and improves the running times of MCM and All-Cavity-MCM problems in the unbalanced case where n ≪ m. We implemented the algorithm (in all modes mentioned above) as a graphical software tool which computes and displays similarities between secondary structures of RNA given as input, and employed it to a preliminary experiment in which we ran all-against-all inter-family pairwise alignments of RNAse P and Hammerhead RNA family members, exposing new similarities which could not be detected by the traditional rooted ordered alignment approaches. The results demonstrate that our approach can be used to expose structural similarity between some RNAs with higher sensitivity than the traditional rooted ordered alignment approaches. Source code and web-interface for our tool can be found in http://www.cs.bgu.ac.il/\~negevcb/FRUUT.