In this paper we explore some implications of view-ing graphs as geometric objects. This approach of-fers 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 the met-ric of the (possibly weighted) graph. Given a graph G 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 small distortion, i.e., make sure that distances between vertices in G closely match the dis-tances between their geometric images. In this paper we develop efficient algorithms for em-bedding graphs low-dimensionally with a small distor-tion. Further algorithmic applications include: 0 A simple, unified approach to a number of prob-lems on multicommodity flows, including the Leighton-Rae Theorem [29] and some of its ex-tensions. 0 For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way, small balanced separators can be found efficiently. Faithful low-dimensional representations of statisti-cal data allow for meaningful and efficient cluster-ing, which is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used

Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 De-randomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 De-randomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Consider N particles, which merge into clusters according to the rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x; y)=N , where K is a specified rate kernel. This MarcusLushnikov model of stochastic coalescence, and the underlying deterministic approximation given by the Smoluchowski coagulation equations, have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x; y) = 1 and K(x; y) = xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously, so many interesting open problems appear. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS96-22859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...

We propose a new randomized algorithm for maintaining a set of clusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an ad-hoc network to collaboratively perform some tasks. 1

In the following, an overview is given on statistical and probabilistic properties of words, as occurring in the analysis of biological sequences. Counts of occurrence, counts of clumps, and renewal counts are distinguished, and exact distributions as well as normal approximations, Poisson process approximations, and compound Poisson approximations are derived. Here, a sequence is modelled as a stationary ergodic Markov chain; a test for determining the appropriate order of the Markov chain is described. The convergence results take the error made by estimating the Markovian transition probabilities into account. The main tools involved are moment generating functions, martingales, Stein’s method, and the Chen-Stein method. Similar results are given for occurrences of multiple patterns, and, as an example, the problem of unique recoverability of a sequence from SBH chip data is discussed. Special emphasis lies on disentangling the complicated dependence structure between word occurrences, due to self-overlap as well as due to overlap between words. The results can be used to derive approximate, and conservative, con � dence intervals for tests. Key words: word counts, renewal counts, Markov model, exact distribution, normal approximation, Poisson process approximation, compound Poisson approximation, occurrences of multiple words, sequencing by hybridization, martingales, moment generating functions, Stein’s method, Chen-Stein method. 1.

The subject matter of this paper is the solution of the linear differential equation y = a#t#y, y#0# = y0 , where y0 2 G, a# # #:R !gand g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus [16], we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.

Test access mechanisms (TAMs) and test wrappers are integral parts of a system-on-chip (SOC) test architecture. Prior research has concentrated on only one aspect of the TAM/wrapper design problem at a time, i.e., either optimizing the TAMs for a set of pre-designed wrappers, or optimizing the wrapper for a given TAM width. In this paper, we address a more general problem, that of carrying out TAM design and wrapper optimization in conjunction. We present an efficient algorithm to construct wrappers that reduce the testing time for cores. Our wrapper design algorithm improves on earlier approaches by also reducing the TAM width required to achieve these lower testing times. We present new mathematical models for TAM optimization that use the core testing time values calculated by our wrapper design algorithm. We further present a new enumerative method for TAM optimization that reduces execution time significantly when the number of TAMs being designed is small. Experimental results are presented for an academic SOC as well as an industrial SOC.

Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)ind-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d-th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.

Abstract — The problem of error-control in a “noncoherent” random network coding channel is considered. Information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V ∩ U is large enough. When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph. Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes. A Reed-Solomonlike code construction is provided and a decoding algorithm given. I.

Abstract. A broadcast encryption system allows a center to communi-cate securely over a broadcast channel with selected sets of users. Each time the set of privileged users changes, the center enacts a protocol to establish a new broadcast key that only the privileged users can obtain, and subsequent transmissions by the center are encrypted using the new broadcast key. We study the inherent trade-off between the number of establishment keys held by each user and the number of transmissions needed to establish a new broadcast key. For every given upper bound on the number of establishment keys held by each user, we prove a lower bound on the number of transmissions needed to establish a new broad-cast key. We show that these bounds are essentially tight, by describing broadcast encryption systems that come close to these bounds. 1