Largest Common Subtree Research Articles

The maximum agreement subtree problem

In this paper, we investigate an extremal problem on binary phylogenetic trees. Given two such trees T1 and T2, both with leaf-set {1,2,…,n}, we are interested in the size of the largest subset S⊆{1,2,…,n} of leaves in a common subtree of T1 and T2. We show that any two binary phylogenetic trees have a common subtree on Ω(logn) leaves, thus improving on the previously known bound of Ω(loglogn) due to Steel and Székely. To achieve this improved bound, we first consider two special cases of the problem: when one of the trees is balanced or a caterpillar, we show that the largest common subtree has Ω(logn) leaves. We then handle the general case by proving and applying a Ramsey-type result: that every binary tree contains either a large balanced subtree or a large caterpillar. We also show that there are constants c,α>0 such that, when both trees are balanced, they have a common subtree on cnα leaves. We conjecture that it is possible to take α=1/2 in the unrooted case, and both c=1 and α=1/2 in the rooted case.

An algebraic view of the relation between largest common subtrees and smallest common supertrees

The relationship between two important problems in tree pattern matching, the largest common subtree and the smallest common supertree problems, is established by means of simple constructions, which allow one to obtain a largest common subtree of two trees from a smallest common supertree of them, and vice versa. These constructions are the same for isomorphic, homeomorphic, topological, and minor embeddings, they take only time linear in the size of the trees, and they turn out to have a clear algebraic meaning.

FAST ALGORITHMS FOR COMPARISON OF SIMILAR UNORDERED TREES

We present fast algorithms for computing the largest common subtree (LCST) and for computing the optimal alignment when two similar unordered trees are given. For the LCST problem for rooted trees, we present an O(4Kn) time algorithm, where n is the maximum size of two input trees and the total number of non-common nodes is bounded by K. We extend this algorithm for unrooted trees and obtain an O(K4Kn) time algorithm. Both of these are subquadratic for two similar trees within K = o( log 4 n) differences. We also show that the alignment problem for rooted and unordered trees of bounded degree can be solved in O(γKn) time for a constant γ. Particularly γ is at most 4.45 for unlabeled binary trees.

Numerical similarity and dissimilarity measures between two trees

Quantifying the measure of similarity between two trees is a problem of intrinsic importance in the study of algorithms and data structures and has applications in computational molecular biology, structural/syntactic pattern recognition and in data management. In this paper we define and formulate an abstract measure of comparison, /spl Omega/(T/sub 1/, T/sub 2/), between two trees T/sub 1/ and T/sub 2/ presented in terms of a set of elementary intersymbol measures /spl omega/(.,.) and two abstract operators /spl oplus/ and /spl otimes/. By appropriately choosing the concrete values for these two operators and for /spl omega/(.,.), this measure can be used to define various quantities including: (1) the edit distance between two trees, (2) the size of their largest common subtree, (3) Prob(T/sub 2/|T/sub 1/), the probability of receiving T/sub 2/ given that T/sub 1/ was transmitted across a channel causing independent substitution and deletion errors, and (4) the a posteriori probability of T/sub 1/ being the transmitted tree given that T/sub 2/ is the received tree containing independent substitution, insertion and deletion errors. The recursive properties of /spl Omega/(T/sub 1/, T/sub 2/) have been derived and a single generic iterative dynamic programming scheme to compute all the above quantities has been developed. The time and space complexities of the algorithm have been analyzed and the implications of our results in both theoretical and applied fields has been discussed.