A linked risk group model for investigating the spread of HIV

This paper describes a model that simulates the spread of HIV and progression to AIDS. The model is based on classical models of disease transmission. It consists of six linked risk groups and tracks the numbers of infectives, AIDS cases, AIDS related deaths, and other deaths of infected persons in each risk group. Parametric functions are used to represent risk-group-specific and time-dependent average contact rates. Contacts are needle sharing, sexual contacts, or blood product transfers.

An important feature of the model is that the contact rate parameters are estimated by minimizing differences between AIDS incidence and reported AIDS cases adjusted for undercounting biases. This feature results in an HIV epidemic curve that is analogous to one estimated by backcalculation models but whose dynamics are determined by simulating disease transmission. The model exhibits characteristics of both the disease transmission and the backcalculation approaches, i.e., the model:

• reconstructs the historical behavior patterns of the different risk groups,

• includes separate effects of treatment and changes in average contact rates,

• accounts for other mortality risks for persons infected with HIV,

• calculates short-term projections of AIDS incidence, HIV incidence, and HIV prevalence,

• calculates cumulative HIV infections (the quantity calculated by backcalculation approaches) and HIV prevalence (the quantity measured by seroprevalence and sentinel surveys). This latter feature permits the validation of the estimates generated by two distinct approaches.

We demonstrate the use of the model with an application to U.S. AIDS data through 1991.     

On small homotopies of loops

Two natural questions are answered in the negative:

• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”

• “Can adding arcs to a space cause an essential curve to become nulhomotopic?”

The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π₁-shape injective.

Area-efficient algorithms for straight-line tree drawings

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.

Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(nlogn) (Crescenzi et al., 1992) to O(nloglogn). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.

Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(nloglogn). If T is binary, we can obtain an O(nloglogn)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/logn) bends in total. Secondly, we present O(nloglogn)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.     

Bregman Voronoi Diagrams

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.     

Randomized incremental construction of simple abstract Voronoi diagrams in 3-space

We introduce the simple abstract Voronoi diagram in 3-space as an abstraction of the usual Voronoi diagram. We show that the 3-dimensional simple abstract Voronoi diagram of n sites can be computed in O(n²) expected time using O(n²) expected space by a randomized algorithm. The algorithm is based on the randomized incremental construction technique of Clarkson and Shor (1989). We apply the algorithm to some concrete types of such diagrams: power diagrams, diagrams under ellipsoid convex distance functions, and diagrams under the Hausdorff distance for sites that are parallel segments all having the same length.     

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain “dominance” map. Asano, Matou?ek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et?al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.     

Computing Voronoi skeletons of a 3-D polyhedron by space subdivision

We tackle the problem of computing the Voronoi diagram of a 3-D polyhedron whose faces are planar. The main difficulty with the computation is that the diagram's edges and vertices are of relatively high algebraic degrees. As a result, previous approaches to the problem have been non-robust, difficult to implement, or not provenly correct.

We introduce three new proximity skeletons related to the Voronoi diagram: (1) the Voronoi graph (VG), which contains the complete symbolic information of the Voronoi diagram without containing any geometry; (2) the approximate Voronoi graph (AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the proximity structure diagram (PSD), which enhances the VG with a geometric approximation of Voronoi elements to any desired accuracy. The new skeletons are important for both theoretical and practical reasons. Many applications that extract the proximity information of the object from its Voronoi diagram can use the Voronoi graphs or the proximity structure diagram instead. In addition, the skeletons can be used as initial structures for a robust and efficient global or local computation of the Voronoi diagram.

We present a space subdivision algorithm to construct the new skeletons, having three main advantages. First, it solves at most uni-variate quartic polynomials. This stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-variate system of equations. Second, the algorithm enables purely local computation of the skeletons in any limited region of interest. Third, the algorithm is simple to implement.     

On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes

It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n²). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n).     

Simplified Voronoi diagrams

We are interested in Voronoi diagrams as a tool in robot path planning, where the search for a path in anr-dimensional space may be simplified to a search on an (r–1)-dimensional Voronoi diagram. We define a Voronoi diagramV based on a measure of distance which is not a true metric. This formulation has lower algebraic complexity than the usual definition, which is a considerable advantage in motion-planning problems with many degrees of freedom. In its simplest form, the measure of distance between a point and a polytope is the maximum of the distances of the point from the half-spaces which pass through faces of the polytope. More generally, the measure is defined in configuration spaces which represent rotation. The Voronoi diagram defined using this distance measure is no longer a strong deformation retract of free space, but it has the following useful property: any path through free space which starts and ends on the diagram can be continuously deformed so that it lies entirely on the diagram. Thus it is still complete for motion planning, but it has lower algebraic complexity than a diagram based on the Euclidean metric.This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's Artificial Intelligence research is provided in part by the Office of Naval Research under Office of Naval Research Contract N00014-81-K-0494 and in part by the Advanced Research Projects Agency under Office of Naval Research Contracts N00014-85-K-0124 and N00014-82-K-0334. John Canny was supported by an IBM fellowship. Bruce Donald was funded in part by a NASA fellowship administered by the Jet Propulsion Laboratory.     

Voronoi diagrams and arrangements

We propose a uniform and general framework for defining and dealing with Voronoi diagrams. In this framework a Voronoi diagram is a partition of a domainD induced by a finite number of real valued functions onD. Valuable insight can be gained when one considers how these real valued functions partitionD ×R. With this view it turns out that the standard Euclidean Voronoi diagram of point sets inR ^d along with its order-k generalizations are intimately related to certain arrangements of hyperplanes. This fact can be used to obtain new Voronoi diagram algorithms. We also discuss how the formalism of arrangements can be used to solve certain intersection and union problems.     

Stable rank boundary principle

Let K denote a compact subset of the complex plane . We present correct proof that the stable rank of A(K) is one. Hereby, A (K) is the algebra of all continuous functions on K which are analytic in the interior of K.

Let G denote a plane domain whose boundary consists of finitely many closed, nonintersecting Jordan curves. We show that for a fixed function of gεC( ), g≠0, the following assertions are equivalent:

Every unimodular element (f, g) is reducible to the principal component exp(C( )).

The zero set Z_g is polynomially convex, i.e., its complement Z_g is connected.

Three problems about simple polygons

We give three related algorithmic results concerning a simple polygon P:

1. Improving a series of previous work, we show how to find a largest pair of disjoint congruent disks inside P in linear expected time.

2. As a subroutine for the above result, we show how to find the convex hull of any given subset of the vertices of P in linear worst-case time.

3. More generally, we show how to compute a triangulation of any given subset of the vertices or edges of P in almost linear time.

A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams

We describe an algorithm for the construction of discretized Voronoi diagrams on a CPU within the context of a large scale numerical fluid simulation. The Discrete Voronoi Chain (DVC) algorithm is fast, flexible and robust. The algorithm stores the Voronoi diagram on a grid or lattice that may be structured or unstructured. The Voronoi diagram is computed by alternatively updating two lists of grid cells per particle to propagate a growth boundary of cells from the particle locations. Distance tests only occur when growth boundaries from different particles collide with each other, hence the number of distance tests is effectively minimized. We give some empirical results for two and three dimensions. The method generalizes to any dimension in a straight forward manner. The distance tests can be based any metric.     

The Voronoi Diagram of Curved Objects

Voronoi diagrams of curved objects can show certain phenomena that are often considered artifacts: The Voronoi diagram is not connected; there are pairs of objects whose bisector is a closed curve or even a two-dimensional object; there are Voronoi edges between different parts of the same site (so-called self-Voronoi-edges); these self-Voronoi-edges may end at seemingly arbitrary points not on a site, and, in the case of a circular site, even degenerate to a single isolated point. We give a systematic study of these phenomena, characterizing their differential-geometric and topological properties. We show how a given set of curves can be refined such that the resulting curves define a “well-behaved” Voronoi diagram. We also give a randomized incremental algorithm to compute this diagram. The expected running time of this algorithm is O(n log n).     

Converting suffix trees into factor/suffix oracles

Several methods to compress suffix trees were defined, most of them with the aim of obtaining compact (that is, space economical) index structures. Besides this practical aspect, a compression method can reveal structural properties of the resulting data structure, allowing a better understanding of it and a better estimation of its performances.

In this paper, we propose a simple method to compress suffix trees by merging couples of nodes. This idea was already used in the literature in a context different from ours. The originality of our approach is that the nodes we merge are not chosen with respect to their subtrees (which is difficult to test algorithmically), nor with respect to the words spelled along branches (which usually requires testing several branches before finding the good one) but with respect to their position in the tree (which is easy to compute). Another particularity of our method is it needs to read no edge label: it is exclusively based on the topology of the suffix tree. The compact structure resulting after compression is the factor/suffix oracle introduced by Allauzen, Crochemore and Raffinot whose accepted language includes the accepted language of the corresponding suffix tree.

The interest of our paper is therefore threefold:

1. A topology-based compression method is defined for (compact) suffix trees.

2. A new property of a factor/suffix oracle is established, that is, like a DAG, it results from the corresponding suffix tree after a linear number of appropriate node mergings; unlike a DAG, the merged nodes do not necessarily have isomorphical subtrees.

3. A new algorithm to transform a suffix tree into a factor/suffix oracle is given, which has linear running time and thus improves the quadratic complexity previously known for the same task.

Link distance and shortest path problems in the plane

This paper describes algorithms to compute Voronoi diagrams, shortest path maps, the Hausdorff distance, and the Fréchet distance in the plane with polygonal obstacles. The underlying distance measures for these algorithms are either shortest path distances or link distances. The link distance between a pair of points is the minimum number of edges needed to connect the two points with a polygonal path that avoids a set of obstacles. The motivation for minimizing the number of edges on a path comes from robotic motions and wireless communications because turns are more difficult in these settings than straight movements.Link-based Voronoi diagrams are different from traditional Voronoi diagrams because a query point in the interior of a Voronoi face can have multiple nearest sites. Our site-based Voronoi diagram ensures that all points in a face have the same set of nearest sites. Our distance-based Voronoi diagram ensures that all points in a face have the same distance to a nearest site.The shortest path maps in this paper support queries from any source point on a fixed line segment. This is a middle-ground approach because traditional shortest path maps typically support queries from either a fixed point or from all possible points in the plane.The Hausdorff distance and Fréchet distance are fundamental similarity metrics for shape matching. This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.     

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L₁-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(nlogn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.

We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.     

Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions

The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R ^d , the maximum complexity of its Voronoi diagram under the L _∞ metric, and also under a simplicial distance function, are both shown to be . The upper bound for the case of the L _∞ metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R ^d . This complexity is , for d ≥ 1 , and it improves to , for d ≥ 2 , if all the hypercubes have the same size. Under the L ₁ metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R ³ is shown to be . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in R ^d under a simplicial or L _∞ distance function. Their expected randomized complexities are for simplicial diagrams and for L _∞ -diagrams. Received July 31, 1995, and in revised form September 9, 1997.     

Unit-distance graphs in Minkowski metric spaces

The Unit-Distance Graph problem in Euclidean plane asks for the minimum number of colors, so that each point on the Euclidean plane can be assigned a single color with the condition that the points at unit distance apart are assigned different colors. It is well known that this number is between 4 and 7, but the exact value is not known. Here this problem is generalized to Minkowski metric spaces and once again the answer is shown to be between 4 and 7. In extreme special cases where the unit circle is a parallelogram or a hexagon the answer is shown to be exactly 4.     

2D Centroidal Voronoi Tessellations with Constraints

<正>We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation.We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges.The clipping itself is efficiently computed by identifying for each constrained edge the(connected) set of triangles whose dual Voronoi vertices are hidden by the constraint.The resulting construction is amenable to Lloyd relaxation so as to obtain a centroidal tessellation with constraints.