Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology

This paper discusses how to count and generate strings that are ``distinct' in two senses: p -distinct and b -distinct. Two strings x on alphabet A and x' on alphabet A' are said to be p -distinct iff they represent distinct ``patterns'; that is, iff there exists no one—one mapping from A to A' that transforms x into x' . Thus aab and baa are p -distinct while aab and ddc are p -equivalent. On the other hand, x and x' are said to be b -distinct iff they give rise to distinct border (failure function) arrays: thus aab with border array 010 is b -distinct from aba with border array 001 . The number of p -distinct (resp. b -distinct) strings of length n formed using exactly k different letters is the [k,n] entry in an infinite p' (resp. b' ) array. Column sums p[n] and b[n] in these arrays give the number of distinct strings of length n . We present algorithms to compute, in constant time per string, all p -distinct (resp. b -distinct) strings of length n formed using exactly k letters, and we also show how to compute all elements p'[k,n] and b'[k,n] . These ideas and results have application to the efficient generation of appropriate test data sets for many string algorithms. Received December 21, 1995; revised April 28, 1997.     

High-Performance Algorithm Engineering for Computational Phylogenetics

A phylogeny is the evolutionary history of a group of organisms; systematists (and other biologists) attempt to reconstruct this history from various forms of data about contemporary organisms. Phylogeny reconstruction is a crucial step in the understanding of evolution as well as an important tool in biological, pharmaceutical, and medical research. Phylogeny reconstruction from molecular data is very difficult: almost all optimization models give rise to NP-hard (and thus computationally intractable) problems. Yet approximations must be of very high quality in order to avoid outright biological nonsense. Thus many biologists have been willing to run farms of processors for many months in order to analyze just one dataset. High-performance algorithm engineering offers a battery of tools that can reduce, sometimes spectacularly, the running time of existing phylogenetic algorithms, as well as help designers produce better algorithms. We present an overview of algorithm engineering techniques, illustrating them with an application to the breakpoint analysis method of Sankoff et al., which resulted in the GRAPPA software suite. GRAPPA demonstrated a speedup in running time by over eight orders of magnitude over the original implementation on a variety of real and simulated datasets. We show how these algorithmic engineering techniques are directly applicable to a large variety of challenging combinatorial problems in computational biology.     

Engineering in genomics-combinatorial optimization in biology

The author suggests that the confidence which many biologists have in problem-solving methods is unwarranted and that there are very important limitations in almost all current methods for solving biological problems. The standard problem solving approach that computer scientists use is outlined. An example of an error in an evolutionary tree problem-the case of the African Eve-is discussed     

Toward new software for computational phylogenetics

Systematists study how a group of genes or organisms evolved. These biologists now have set their sights on the Tree of Life challenge: to reconstruct the evolutionary history of all known living organisms. A typical phylogenetic reconstruction starts with biomolecular data, such as DNA sequences for modern organisms, and builds a tree, or phylogeny, for these sequences that represents a hypothesized evolutionary history. Finding the best tree for a data set can be a computationally intensive problem. Phylogenetic software for mapping the Tree of Life will require entirely new approaches in statistical models of evolution, high-performance implementations, and data analysis and visualization. The authors have developed polynomial algorithmic techniques for use in their research addressing phylogeny reconstruction from biomolecular sequences, focusing on the accuracy of reconstructions and the use of simulations     

A robust model for finding optimal evolutionary trees

Constructing evolutionary trees for species sets is a fundamental problem in computational biology. One of the standard models assumes the ability to compute distances between every pair of species, and seeks to find an edge-weighted treeT in which the distanced _ij ^T in the tree between the leaves ofT corresponding to the speciesi andj exactly equals the observed distance,d _ij . When such a tree exists, this is expressed in the biological literature by saying that the distance function or matrix isadditive, and trees can be constructed from additive distance matrices in0(n ²) time. Real distance data is hardly ever additive, and we therefore need ways of modeling the problem of finding the best-fit tree as an optimization problem.In this paper we present several natural and realistic ways of modeling the inaccuracies in the distance data. In one model we assume that we have upper and lower bounds for the distances between pairs of species and try to find an additive distance matrix between these bounds. In a second model we are given a partial matrix and asked to find if we can fill in the unspecified entries in order to make the entire matrix additive. For both of these models we also consider a more restrictive problem of finding a matrix that fits a tree which is not only additive but alsoultrametric. Ultrametric matrices correspond to trees which can be rooted so that the distance from the root to any leaf is the same. Ultrametric matrices are desirable in biology since the edge weights then indicate evolutionary time. We give polynomial-time algorithms for some of the problems while showing others to be NP-complete. We also consider various ways of fitting a given distance matrix (or a pair of upper- and lower-bound matrices) to a tree in order to minimize various criteria of error in the fit. For most criteria this optimization problem turns out to be NP-hard, while we do get polynomial-time algorithms for some.Supported by DIMACS under NSF Contract STC-88-09648.Supported by NSF Grant CCR-9108969.This work was begun while this author was visiting DIMACS in July and August 1992, and was supported in part by the U.S. Department of Energy under Contract DE-AC04-76DP00789.