Showing papers by "Yuri Rabinovich published in 1995"

The geometry of graphs and some of its algorithmic applications

Abstract: In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Given a graphG we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a smalldistortion, i.e., make sure that distances between vertices inG closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include: Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficientclustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) ann-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(logn). This result is tight, and closes a gap left open by Bourgain [12].

A computational view of population genetics

Abstract: A Computational View of Population Genetics (preliminary version) Yuval Rabanit Yuri Rabinovicht Alistair Sinclair] This paper contributes to the study of nonlinear dynamical systems from a computational perspective. These systems are inherently more powerful than their linear counterparts (such as Markov chains), which have had a wide impact in Computer Science, and they seem likely to play an increasing role in future. However, there are as yet no general techniques available for handling the computational aspects of discrete nonlinear systems, and even the simplest examples seem very hard to analyze. We focus in this paper on a class of quadratic systems that are widely used as a model in population genetics and also in genetic algorithms. These systems describe a process where random matings occur between parental chromosomes via a mechanism known as “crossover”: i.e., children inherit pieces of genetic material from different parents according to some random rule. Our results concern two fundamental quantitative properties of crossover systems: 1. We develop a general technique for computing the rate of convergence to equilibrium. We apply this technique to obtain tight bounds on the rate of convergence in several cases of biological and computational interest. In general, we prove that these systems are “rapidly mixing”, in the sense that the convergence time is very small in comparison with the size of the state space. 2. We show that, for crossover systems, the classical quadratic system is a good model for the behavior of finite populations of small size. This stands in sharp contrast to recent results of Arora et al and Pudlak, who show that such a correspondence is unlikely to hold for general quadratic systems. tD~p~~t~ent Of Computer Science, University of TorontO, Toronto, Ontario M5S 1A4, Canada. Email: {rabani ,yuri}Q cs. toronto. edu. t Computer Science Division, University of California, Berkeley CA 94720-1776, U.S.A. Email: sinclair@cs .berkeley. edu. Perrmssion to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyri ht notice and the % title of the publication and Its dat~ appear, an notice IS gwen that copyin is by permission of tne Association of Computing ? Machinery o copy otherwise, or to republish, requires a fee andlor specific permission. STOC” 95, Las Vegas, Nevada, USA @ 1995 ACM 0-89791 -718-9/95/0005 .$3.50