Rephael Wenger

Embedding planar graphs at fixed vertex locations

Abstract. Let G be a planar graph of n vertices, v1,…,vn, and let {p1,…,pn} be a set of n points in the plane. We present an algorithm for constructing in O(n2) time a planar embedding of G, where vertex vi is represented by point pi and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/403 bends.

Geometric Transversal Theory

Geometric transversal theory has its origins in Helly’s theorem: Theorem 1.1 (Helly’s Theorem) [49]. Suppose A is a family of at least d + 1 convex sets in IR d , and A is finite or each member of A is compact. Then if every d + 1 members of A have a common point, there is a point common to all the members of A.

Extremal graphs with no C 4, s, or C 10, s

We present a new construction for arbitrarily large graphs with n vertices, (n2)32 edges and no C4, (n2)43 edges and no C6, and (n2)65 edges and no C10.

Isosurfacing in higher dimensions

Visualization algorithms have seen substantial improvements in the past several years. However, very few algorithms have been developed for directly studying data in dimensions higher than three. Most algorithms require a sampling in three-dimensions before applying any visualization algorithms. This sampling typically ignores vital features that may be present when examined in oblique cross-sections, and places an undo burden on system resources when animation through additional dimensions is desired. For time-varying data of large data sets, smooth animation is desired at interactive rates. We provide a fast Marching Cubes like algorithm for hypercubes of any dimension. To support this, we have developed a new algorithm to automatically generate the isosurface and triangulation tables for any dimension. This allows the efficient calculation of 4D isosurfaces, which can be interactively sliced to provide smooth animation or slicing through oblique hyperplanes. The former allows for smooth animation in a very compressed format. The latter provide better tools to study time-evolving features as they move downstream. We also provide examples in using this technique to show interval volumes or the sensitivity of a particular isovalue threshold.

Isosurface construction in any dimension using convex hulls

We present an algorithm for constructing isosurfaces in any dimension. The input to the algorithm is a set of scalar values in a d-dimensional regular grid of (topological) hypercubes. The output is a set of (d-1)-dimensional simplices forming a piecewise linear approximation to the isosurface. The algorithm constructs the isosurface piecewise within each hypercube in the grid using the convex hull of an appropriate set of points. We prove that our algorithm correctly produces a triangulation of a (d-1 )-manifold with boundary. In dimensions three and four, lookup tables with 2/sup 8/ and 2/sup 16/ entries, respectively, can be used to speed the algorithms running time. In three dimensions, this gives the popular marching cubes algorithm. We discuss applications of four-dimensional isosurface construction to time varying isosurfaces, interval volumes, and morphing.

Volume tracking using higher dimensional isosurfacing

Tracking and visualizing local features from a time-varying volumetric data allows the user to focus on selected regions of interest, both in space and time, which can lead to a better understanding of the underlying dynamics. In this paper, we present an efficient algorithm to track time-varying isosurfaces and interval volumes using isosurfacing in higher dimensions. Instead of extracting the data features such as isosurfaces or interval volumes separately from multiple time steps and computing the spatial correspondence between those features, our algorithm extracts the correspondence directly from the higher dimensional geometry and thus can more efficiently follow the user selected local features in time. In addition, by analyzing the resulting higher dimensional geometry, it becomes easier to detect important topological events and the corresponding critical time steps for the selected features. With our algorithm, the user can interact with the underlying time-varying data more easily. The computation cost for performing time-varying volume tracking is also minimized.

DNA copy number gains in head and neck squamous cell carcinoma

Gene amplification, a common mechanism for oncogene activation in cancer, has been used as a tag for the identification of novel oncogenes. DNA amplification is frequently observed in head and neck squamous cell carcinoma (HNSCC) and potential oncogenes have already been reported. We applied restriction landmark genome scanning (RLGS) to study gene amplifications and low-level copy number changes in HNSCC in order to locate previously uncharacterized regions with copy number gains in primary tumor samples. A total of 63 enhanced RLGS fragments, indicative of DNA copy number changes, including gains of single alleles, were scored. Enhanced sequences were identified from 33 different chromosomal regions including those previously reported (e.g. 3q26.3 and 11q13.3) as well as novel regions (e.g. 3q29, 8q13.1, 8q22.3, 9q32, 10q24.32, 14q32.32, 17q25.1 and 20q13.33). Furthermore, our data suggest that amplicons 11q13.3 and 3q26.3–q29 may be divided into possibly two and three independent amplicons, respectively, an observation supported by published microarray expression data.

On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution

This paper addresses the compile-time optimization of a form of nested-loop computation that is motivated by a computational physics application. The computations involve multi-dimensional surface and volume integrals where the integrand is a product of a number of array terms. Besides the issue of optimal distribution of the arrays among the processors, there is also scope for reordering of the operations using the commutativity and associativity properties of addition and multiplication, and the application of the distributive law to significantly reduce the number of operations executed. A formalization of the operation minimization problem and proof of its NP-completeness is provided. A pruning search strategy for determination of an optimal form is developed. An analysis of the communication requirements and a polynomial-time algorithm for determination of optimal distribution of the arrays are also provided.

A randomized O ( m log m ) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

Given a continuous scalar field ƒ: <i>X</i> → where <i>X</i> is a topological space, a <i>level set</i> of ƒ is a set {<i>x</i> ∈ <i>X</i> : ƒ (<i>x</i>) = α} for some value α ∈ IR. The level sets of ƒ can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of ƒ encodes these changes in connected components of level sets. It provides a simple yet meaningful abstraction of the input domain. As such, it has been used in a range of applications in fields such as graphics and scientific visualization. In this paper, we present the first sub-quadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex <i>K</i>. Our algorithm is randomized with an expected running time <i>O</i>(<i>m</i> log <i>n</i>), where <i>m</i> is the size of the 2-skeleton of <i>K</i> (i.e, total number of vertices, edges and triangles), and <i>n</i> is the number of vertices. This presents a significant improvement over the previous Θ(<i>mn</i>) time complexity for arbitrary complex, matches (although in expectation only) the best known result for the special case of 2-manifolds, and is faster than current algorithms for any other special cases (e.g, 3-manifolds). Our algorithm is also very simple to implement. Preliminary experimental results show that it performs well in practice.

Embedding Planar Graphs at Fixed Vertex Locations

Let G be a planar graph of n vertices, v1,..., vn, and let {p1,...,pn} be a set of n points in the plane. We present an algorithm for constructing in O(n2) time a planar embedding of G, where vertex vi is represented by point pi and each edge is represented by a polygonal curve with O(n) bends (internal vertices.) This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/403 bends.