Minimal Phylogenetic Supertrees and Local Consensus Trees

The problem of constructing a minimally resolved phylogenetic supertree (i.e., having the smallest possible number of internal nodes) that contains all of the rooted triplets from a consistent set R is known to be NP-hard. In this paper, we prove that constructing a phylogenetic tree consistent with R that contains the minimum number of additional rooted triplets is also NP-hard, and develop exact, exponential-time algorithms for both problems. The new algorithms are applied to construct two variants of the local consensus tree; for any set S of phylogenetic trees over some leaf label set L, this gives a minimal phylogenetic tree over L that contains every rooted triplet present in all trees in S, where ``minimal\u27\u27 means either having the smallest possible number of internal nodes or the smallest possible number of rooted triplets. The second variant generalizes the RV-II tree, introduced by Kannan, Warnow, and Yooseph in 1998

Fixed Parameter Polynomial Time Algorithms for Maximum Agreement and Compatible Supertrees

Consider a set of labels $L$ and a set of trees {\mathcal T} = \{{\mathcal T}^{(1), {\mathcal T}^{(2), ..., {\mathcal T}^{(k) \$ where each tree {\mathcal T}^{(i)$is distinctly leaf-labeled by some subset of$L$. One fundamental problem is to find the biggest tree (denoted as supertree) to represent$\mathcal T}$which minimizes the disagreements with the trees in${\mathcal T}$under certain criteria. This problem finds applications in phylogenetics, database, and data mining. In this paper, we focus on two particular supertree problems, namely, the maximum agreement supertree problem (MASP) and the maximum compatible supertree problem (MCSP). These two problems are known to be NP-hard for$k \geq 3$. This paper gives the first polynomial time algorithms for both MASP and MCSP when both$k$and the maximum degree$D$ of the trees are constant

On Finding the Adams Consensus Tree

This paper presents a fast algorithm for finding the Adams consensus tree of a set of conflicting phylogenetic trees with identical leaf labels, for the first time improving the time complexity of a widely used algorithm invented by Adams in 1972 [1]. Our algorithm applies the centroid path decomposition technique [9] in a new way to traverse the input trees\u27 centroid paths in unison, and runs in O(k n log n) time, where k is the number of input trees and n is the size of the leaf label set. (In comparison, the old algorithm from 1972 has a worst-case running time of O(k n^2).) For the special case of k = 2, an even faster algorithm running in O(n cdot frac{log n}{loglog n}) time is provided, which relies on an extension of the wavelet tree-based technique by Bose et al. [6] for orthogonal range counting on a grid. Our extended wavelet tree data structure also supports truncated range maximum queries efficiently and may be of independent interest to algorithm designers

A Decomposition Theorem for Maximum Weight Bipartite Matchings

Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N and W be the node count, the largest edge weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G-{u} for all nodes u in O(W) time.Comment: The journal version will appear in SIAM Journal on Computing. The conference version appeared in ESA 199

An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm.Comment: To appear in Journal of Algorithm

Cavity Matchings, Label Compressions, and Unrooted Evolutionary Trees

We present an algorithm for computing a maximum agreement subtree of two unrooted evolutionary trees. It takes O(n^{1.5} log n) time for trees with unbounded degrees, matching the best known time complexity for the rooted case. Our algorithm allows the input trees to be mixed trees, i.e., trees that may contain directed and undirected edges at the same time. Our algorithm adopts a recursive strategy exploiting a technique called label compression. The backbone of this technique is an algorithm that computes the maximum weight matchings over many subgraphs of a bipartite graph as fast as it takes to compute a single matching

Using indirect protein interactions for the prediction of Gene Ontology functions

10.1186/1471-2105-8-S4-S8BMC Bioinformatics8SUPPL. 4BBMI