Subtree Problem Research Articles

On the complexity of comparing evolutionary trees

We study the computational complexity and approximation of several problems arising in the comparison of evolutionary trees. It is shown that the maximum agreement subtree (MAST) problem for three trees with unbounded degree cannot be approximated within ratio 2 log δ n in polynomial time for any δ < 1, unless NP ⊆ DTIME[2 polylog n ], and MAST with edge contractions for two binary trees is NP-hard. This answers two open questions posed in [1]. For the maximum refinement subtree (MRST) problem involving two trees, we show that it is polynomial-time solvable when both trees have bounded degree and is NP-hard when one of the trees can have an arbitrary degree. Finally, we consider the problem of optimally transforming a tree into another by transferring subtrees around. It is shown that computing the subtree-transfer distance is NP-hard and an approximation algorithm with performance ratio 3 is given.

Optimizing constrained subtrees of trees

Given a tree G = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the ''rooted subtree problem'', is to find a maximum weight subtree, with a specified root, from a given set of subtrees. The second problem, called ''the subtree packing problem'', is to find a maximum weight packing of node disjoint subtrees chosen from a given set of subtrees, where the value of each subtree may depend on its root. We show that the complexity status of both problems is related, and that the subtree packing problem is polynomial if and only if each rooted subtree problem is polynomial, In addition we show that the convex hulls of the feasible solutions to both problems are related: the convex hull of solutions to the packing problem is given by ''pasting together'' the convex hulls of the rooted subtree problems. We examine in detail the case where the set of feasible subtrees rooted at node i consists of all subtrees with at most k nodes, For this case we derive valid inequalities, and specify the convex hull when k less than or equal to 4.

Fast Comparison of Evolutionary Trees

Constructing evolutionary trees for species sets is a fundamental problem in biology. Unfortunately, there is no single agreed upon method for this task, and many methods are in use. Current practice dictates that trees be constructed using different methods and that the resulting trees then be compared for consensus. It has become necessary to automate this process as the number of species under consideration has grown. We study the Unrooted Maximum Agreement Subtree Problem ( UMAST ) and its rooted variant ( RMAST ). The UMAST problem is as follows: given a set A and two trees T 0 and T 1 leaf-labeled by the elements of A , find a maximum cardinality subset B of A such that the restrictions of T 0 and T 1 to B are topologically isomorphic. Our main result is an O ( n 2 + o (1) ) time algorithm for the UMAST problem. We also derive an O ( n 2 ) time algorithm for the RMAST problem. The previous best algorithm for both these problems has running time O ( n 4.5 + o (1) ).

A simplified graphical method for deriving system steady-state probability

This paper presents an interesting relation between the Maximum Agreement Subtree (MAST) problem and the Tree Edit Distance (TED) problem, both of which have been intensively studied in the literature. To be specific, we show that, for an arbitrary tree edit distance metric that is a derivative of the Taï tree edit distance metric, there exists a MAST-like pattern extraction problem, named Mostly Adjusted Sub-Forest (MASF) problem, such that computing a distance between trees x and y is equivalent to finding an optimal pattern shared between x and y. The MASF problem is different from the MAST problem in: (1) A pattern of the MASF problem may be a forest instead of a tree; (2) Instead of requiring exact match of labels, a pattern of a MASF problem is multi-labeled, and our flexible matching rule only requires that the label set of a vertex of a pattern includes the label of the corresponding vertex in a target tree; (3) To control ambiguity of matching caused by the flexible rule, the objective function includes a penalty function. As an application of this generic framework, we equate the Lu* tree edit distance metric with a pattern extraction problem, named the Mostly Adjusted Agreement Sub-Tree (MAAST) problem. The MAAST problem aims to find optimal agreement subtrees under the flexible matching rule and is solved in O(n2dlog⁡d)-time, where n and d are the size and the minimum degree of the input trees. The MAST problem requires O(n2.5)-time computation.

Adaptive tree-structured self generating radial basis function

Abstract In this paper, we propose a new identification method called “Adaptive Tree-Structured Self Generating Radial Basis Function(ATree-RBF) ”. In this method, we combine the Sanger's tree-structured adaptive network for overall model structure, with the Maximum Absolute Error(MAE) selection method for sub-tree identification problem.

Kaikoura tree theorems: Computing the maximum agreement subtree

The Maximum Agreement Subtree Problem was posed by Finden and Gordon in 1985, and is as follows: given a set S = { S 1, S 2,…, S n } and two trees P and Q leaf-labelled by the elements of S, find a maximum cardinality subset S 0 of S such that P| S 0 = Q| S 0. This problem arises in evaluationary tree construction, where different methods or data yield (possibly) different trees for the same species on which the trees agree. A superpolynomial time algorithm for finding a maximum agreement subtree of two binary trees was found by Kubicka et al. In this paper, we will present an O( n 4.5 log n + V) algorithm to determine the largest agreement subtree of two trees on n leaves, where V is the maximum number of nodes in the trees. For the case of trees of maximum degree k, there are two algorithms presented: one has running time O( k! n 2 + V) while the other has running time O( k 2.5 n 2 log n + V). These algorithms also apply when the trees are constrained to be rooted; in this case a maximum agreement subtree is also constrained to be rooted. Each of the algorithms we present can be modified to produce a maximum agreement subtree, rather than computing only the size. Thus, we can solve the searchproblem in the same running time as above.

Indirect covering tree problems on spanning tree networks

This paper introduces two types of indirect covering tree problems using a spanning tree as a backbone network. The minimum cost covering subtree seeks the minimum cost collection of arcs that form a subtree and satisfy covering constraints for nodes of the network. Reduction techniques that have been used to solve the location set covering problem are extended to solve the Minimum Cost Covering Subtree (MCCS). The second problem, The Maximal Indirect Covering Subtree (MICS), chooses that subtree which maximizes the demand within a distance standard of nodes of the subtree. In addition, some variations of the maximal indirect covering subtree problem are proposed.

Some comments on the subtree isomorphism problem for ordered trees

We give counterexamples to show that the algorithm presented recently by Makinen fails to solve the subtree isomorphism problem for rooted ordered trees. We also give linear time algorithms for the two versions of the subtree isomorphism problem for rooted ordered trees.

An Analysis of a Good Algorithm for the Subtree Problem, Corrected

It is shown that the proof of the main result in Reyner’s paper, similarly titled, is incorrect. Interestingly, by combining a simple modification of the algorithm with tighter analysis, one can obtain the original result with a minor improvement.

A remark on the time complexity of the subtree problem

In the present paper it is shown that the subtree problem may be solved in the general case in time 0 (n 2.5) and in the case of bounded degree in time 0 (n 2) (n denotes the length of the input).

An Analysis of a Good Algorithm for the Subtree Problem

A good algorithm is analyzed for deciding if one tree is a subtree of another tree. If both trees are rooted, the smaller tree has n vertices and the larger has m vertices; then the total number of computations is $O(nm^{1.5})$ or better, depending on how good an algorithm one has for a maximal matching in a bipartite graph.