Teresa M. Przytycka

A simple parallel tree contraction algorithm

Abstract A simple reduction from the tree contraction problem to the list ranking problem is presented. The reduction takes O(log n) time for a tree with n nodes, using O( n log n ) EREW processors. Thus tree contraction can be done as efficiently as list ranking. A broad class of parallel tree computations to which the tree contraction techniques apply is described. This subsumes earlier characterizations. Applications to the computation of certain properties of cographs are presented in some detail.

On the agreement of many trees

Abstract We consider the problem of computing the Maximum Agreement Subtree (MAST) of a set of rooted leaf labeled trees. We give an algorithm which computes the MAST of k trees on n leaves where some tree has maximum outdegree d in time O( kn 3 + n d ).

A protein taxonomy based on secondary structure

Does a proteins secondary structure determine its three-dimensional fold? This question is tested directly by analyzing proteins of known structure and constructing a taxonomy based solely on secondary structure. The taxonomy is generated automatically, and it takes the form of a tree in which proteins with similar secondary structure occupy neighboring leaves. Our tree is largely in agreement with results from the structural classification of proteins (SCOP), a multidimensional classification based on homologous sequences, full three-dimensional structure, information about chemistry and evolution, and human judgment. Our findings suggest a simple mechanism of protein evolution.

An O ( n log n ) Algorithm for the Maximum Agreement Subtree Problem for Binary Trees

The maximum agreement subtree problem is the following. Given two rooted trees whose leaves are drawn from the same set of items (e.g., species), find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic. We consider the case which occurs frequently in practice, i.e., the case when the trees are binary, and give an O(nlog n) time algorithm for this problem.

On an Optimal Split Tree Problem

We introduce and study a problem that we refer to as the optimal split tree problem. The problem generalizes a number of problems including two classical tree construction problems including the Huffman tree problem and the optimal alphabetic tree. We show that the general split tree problem is NP-complete and analyze a greedy algorithm for its solution. We show that a simple modification of the greedy algorithm guarantees O(log n) approximation ratio. We construct an example for which this algorithm achieves Ω(log n/log log n) approximation ratio. We show that if all weights are equal and the optimal split tree is of depth O(log n). then the greedy algorithm guarantees O(log n/log log n) approximation ratio. We also extend our approximation algorithm to the construction of a search tree for partially ordered sets.

Grid intersection graphs and boxicity

Abstract A graph has boxicity k if k is the smallest integer such that G is an intersection graph of k -dimensional boxes in a k -dimensional space (where the sides of the boxes are parallel to the coordinate axis). A graph has grid dimension k if k is the smallest integer such that G is an intersection graph of k -dimensional boxes (parallel to the coordinate axis) in a ( k + 1)-dimensional space. We prove that all bipartite graphs with boxicity two, have grid dimensions one, that is, they can be represented as intersection graphs of horizontal and vertical intervals in the plane. We also introduce some inequalities for the grid dimension of a graph, and discuss extremal graphs with large grid dimensions.

Recursive domains in proteins

The domain is a fundamental unit of protein structure. Numerous studies have analyzed folding patterns in protein domains of known structure to gain insight into the underlying protein folding process. Are such patterns a haphazard assortment or are they similar to sentences in a language, which can be generated by an underlying grammar? Specifically, can a small number of intuitively sensible rules generate a large class of folds, including feasible new folds? In this paper, we explore the extent to which four simple rules can generate the known all‐β folds, using tools from graph theory. As a control, an exhaustive set of β‐sandwiches was tested and found to be largely incompatible with such a grammar. The existence of a protein grammar has potential implications for both the mechanism of folding and the evolution of domains.

Constructing Huffman Trees in Parallel

We present a parallel algorithm for the Huffman coding problem. We reduce the Huffman coding problem to the concave least weight subsequence problem and give a parallel algorithm that solves the latter problem in

General techniques for comparing unrooted evolutionary trees

O(\sqrt n \log n)

On the Complexity of String Folding

time with