Labeled Interleaving Distance for Reeb Graphs

Merge trees, contour trees, and Reeb graphs are graph-based topological descriptors that capture topological changes of (sub)level sets of scalar fields. Comparing scalar fields using their topological descriptors has many applications in topological data analysis and visualization of scientific data. Recently, Munch and Stefanou introduced a labeled interleaving distance for comparing two labeled merge trees, which enjoys a number of theoretical and algorithmic properties. In particular, the labeled interleaving distance between merge trees can be computed in polynomial time. In this work, we define the labeled interleaving distance for labeled Reeb graphs. We then prove that the (ordinary) interleaving distance between Reeb graphs equals the minimum of the labeled interleaving distance over all labelings. We also provide an efficient algorithm for computing the labeled interleaving distance between two labeled contour trees (which are special types of Reeb graphs that arise from simply-connected domains). In the case of merge trees, the notion of the labeled interleaving distance was used by Gasparovic et al. to prove that the (ordinary) interleaving distance on the set of (unlabeled) merge trees is intrinsic. As our final contribution, we present counterexamples showing that, on the contrary, the (ordinary) interleaving distance on (unlabeled) Reeb graphs (and contour trees) is not intrinsic. It turns out that, under mild conditions on the labelings, the labeled interleaving distance is a metric on isomorphism classes of Reeb graphs, analogous to the ordinary interleaving distance. This provides new metrics on large classes of Reeb graphs

Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals

We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is an intuitive quality measure for drawings of a graph that measures the height of a drawing ϕ : G↪ℝ² as follows. For a vertical line in ℝ², let the height of be the cardinality of the set ∩ ϕ(G). The height of a drawing of G is the maximum height over all vertical lines. In this paper, instead of abstract graphs, we fix a drawing and consider plane graphs. In other words, we are looking for a homeomorphism of the plane that minimizes the height of the resulting drawing. This problem is equivalent to the homotopy height problem in the plane, and the homotopic Fréchet distance problem. These problems were recently shown to lie in NP, but no polynomial-time algorithm or NP-hardness proof has been found since their formulation in 2009. We present the first polynomial-time algorithm for drawing trees with optimal height. This corresponds to a polynomial-time algorithm for the homotopy height where the triangulation has only one vertex (that is, a set of loops incident to a single vertex), so that its dual is a tree

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve ?, and a collection of disjoint normal curves ?, there is a polynomial-time algorithm to decide if ? lies in the normal subgroup generated by components of ? in the fundamental group of the surface after attaching the curves to a basepoint

The Long-Baseline Neutrino Experiment: Exploring Fundamental Symmetries of the Universe

The preponderance of matter over antimatter in the early Universe, the dynamics of the supernova bursts that produced the heavy elements necessary for life and whether protons eventually decay --- these mysteries at the forefront of particle physics and astrophysics are key to understanding the early evolution of our Universe, its current state and its eventual fate. The Long-Baseline Neutrino Experiment (LBNE) represents an extensively developed plan for a world-class experiment dedicated to addressing these questions. LBNE is conceived around three central components: (1) a new, high-intensity neutrino source generated from a megawatt-class proton accelerator at Fermi National Accelerator Laboratory, (2) a near neutrino detector just downstream of the source, and (3) a massive liquid argon time-projection chamber deployed as a far detector deep underground at the Sanford Underground Research Facility. This facility, located at the site of the former Homestake Mine in Lead, South Dakota, is approximately 1,300 km from the neutrino source at Fermilab -- a distance (baseline) that delivers optimal sensitivity to neutrino charge-parity symmetry violation and mass ordering effects. This ambitious yet cost-effective design incorporates scalability and flexibility and can accommodate a variety of upgrades and contributions. With its exceptional combination of experimental configuration, technical capabilities, and potential for transformative discoveries, LBNE promises to be a vital facility for the field of particle physics worldwide, providing physicists from around the globe with opportunities to collaborate in a twenty to thirty year program of exciting science. In this document we provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess.Comment: Major update of previous version. This is the reference document for LBNE science program and current status. Chapters 1, 3, and 9 provide a comprehensive overview of LBNE's scientific objectives, its place in the landscape of neutrino physics worldwide, the technologies it will incorporate and the capabilities it will possess. 288 pages, 116 figure