Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures improved, references update

Reconstructing a Simple Polytope from its Graph

Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertex-facet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Koerner (2001).Comment: 14 page

Cliffbot Maestro

Cliffbot Maestro permits teleoperation of remote rovers for field testing in extreme environments. The application user interface provides two sets of tools for operations: stereo image browsing and command generation

Spatial Query for Planetary Data

Science investigators need to quickly and effectively assess past observations of specific locations on a planetary surface. This innovation involves a location-based search technology that was adapted and applied to planetary science data to support a spatial query capability for mission operations software. High-performance location-based searching requires the use of spatial data structures for database organization. Spatial data structures are designed to organize datasets based on their coordinates in a way that is optimized for location-based retrieval. The particular spatial data structure that was adapted for planetary data search is the R+ tree

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Parallel Eclipse Project Checkout

Parallel Eclipse Project Checkout (PEPC) is a program written to leverage parallelism and to automate the checkout process of plug-ins created in Eclipse RCP (Rich Client Platform). Eclipse plug-ins can be aggregated in a feature project. This innovation digests a feature description (xml file) and automatically checks out all of the plug-ins listed in the feature. This resolves the issue of manually checking out each plug-in required to work on the project. To minimize the amount of time necessary to checkout the plug-ins, this program makes the plug-in checkouts parallel. After parsing the feature, a request to checkout for each plug-in in the feature has been inserted. These requests are handled by a thread pool with a configurable number of threads. By checking out the plug-ins in parallel, the checkout process is streamlined before getting started on the project. For instance, projects that took 30 minutes to checkout now take less than 5 minutes. The effect is especially clear on a Mac, which has a network monitor displaying the bandwidth use. When running the client from a developer s home, the checkout process now saturates the bandwidth in order to get all the plug-ins checked out as fast as possible. For comparison, a checkout process that ranged from 8-200 Kbps from a developer s home is now able to saturate a pipe of 1.3 Mbps, resulting in significantly faster checkouts. Eclipse IDE (integrated development environment) tries to build a project as soon as it is downloaded. As part of another optimization, this innovation programmatically tells Eclipse to stop building while checkouts are happening, which dramatically reduces lock contention and enables plug-ins to continue downloading until all of them finish. Furthermore, the software re-enables automatic building, and forces Eclipse to do a clean build once it finishes checking out all of the plug-ins. This software is fully generic and does not contain any NASA-specific code. It can be applied to any Eclipse-based repository with a similar structure. It also can apply build parameters and preferences automatically at the end of the checkout

Targeting and Localization for Mars Rover Operations

A design and a partially developed application framework were presented for improving localization and targeting for surface spacecraft. The program has value for the Mars Science Laboratory mission, and has been delivered to support the Mars Exploration Rovers as part of the latest version of the Maestro science planning tool. It also has applications for future missions involving either surface-based or low-altitude atmospheric robotic vehicles. The targeting and localization solutions solve the problem of how to integrate localization estimate updates into operational planning tools, operational data product generalizations, and flight software by adding expanded flexibility to flight software, the operations data product pipeline, and operations planning tools based on coordinate frame updates during a planning cycle

On hydrogen bond correlations at high pressures

In situ high pressure neutron diffraction measured lengths of O H and H O pairs in hydrogen bonds in substances are shown to follow the correlation between them established from 0.1 MPa data on different chemical compounds. In particular, the conclusion by Nelmes et al that their high pressure data on ice VIII differ from it is not supported. For compounds in which the O H stretching frequencies red shift under pressure, it is shown that wherever structural data is available, they follow the stretching frequency versus H O (or O O) distance correlation. For compounds displaying blue shifts with pressure an analogy appears to exist with improper hydrogen bonds.Comment: 12 pages,4 figure

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

Optimal topological simplification of discrete functions on surfaces

We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse theory and persistent homology. Our method completely removes homological noise with persistence less than 2{\delta}, constructively proving the tightness of a lower bound on the number of critical points given by the stability theorem of persistent homology in dimension two for any input function. We also show that an optimal solution can be computed in linear time after persistence pairs have been computed.Comment: 27 pages, 8 figure