Motion Planning for Unlabeled Discs with Optimality Guarantees

We study the problem of path planning for unlabeled (indistinguishable) unit-disc robots in a planar environment cluttered with polygonal obstacles. We introduce an algorithm which minimizes the total path length, i.e., the sum of lengths of the individual paths. Our algorithm is guaranteed to find a solution if one exists, or report that none exists otherwise. It runs in time $\tilde{O}(m^4+m^2n^2)$, where $m$ is the number of robots and $n$ is the total complexity of the workspace. Moreover, the total length of the returned solution is at most $\text{OPT}+4m$, where OPT is the optimal solution cost. To the best of our knowledge this is the first algorithm for the problem that has such guarantees. The algorithm has been implemented in an exact manner and we present experimental results that attest to its efficiency

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469

Screen Capture for Sensitive Systems

Maintaining usable security in application domains such as healthcare or power systems requires an ongoing conversation among stakeholders such as end-users, administrators, developers, and policy makers. Each party has power to influence the design and implementation of the application and its security posture, and effective communication among stakeholders is one key to achieving influence and adapting an application to meet evolving needs. In this thesis, we develop a system that combines keyboard/video/mouse (KVM) capture with automatic text redaction to produce precise technical content that can enrich stakeholder communications, improve end-user influence on system evolution, and help reveal the definition of ``usable security.\u27\u27 Text-redacted screen captures reduce sensitivity of captured material and thus can facilitate timely data sharing among stakeholders. KVM-based capture makes our system both application and operating-system independent because it eliminates software-interface dependencies on capture targets. Thus, our work can be used to instrument closed or certified systems where capture software cannot be installed or documentation and support lack. It can instrument widely-varying platforms that lack standards-compliance and interoperability or redact special document formats while displayed onscreen. We present three techniques for redacting text from screenshots and two redaction applications. One application can capture, text redact, and edit screen video and the other can text redact and edit static screenshots. We also present empirical measurements of redaction effectiveness and processing latency to demonstrate system performance. When applied to our principal dataset, redaction removes text with over 93\% accuracy and simultaneously preserves more than 76\% of image pixels on average. Thus by default, it retains more visual context than a technique such as blindly redacting entire screenshots. Finally, our system redacts each screenshot in 0.1 to 21 seconds depending on which technique it applies

Numerical limitations of the attainment of the orientation of geological planes

The paper discusses limitations of analytical attainment of the attitude of a geological plane by using three non-collinear points. We present problems that arise during computing the orientation of a plane generated by almost collinear points. We referred these errors to floating-point arithmetic inaccuracies. To demonstrate the problem, we examined a surface of constant orientation. We used Delaunay triangulation to calculate its local orientation parameters. We introduced a new measure of collinearity applicable for collecting attitude of planar triangles. Using this measure we showed that certain planes generated by the triangulation cannot be treated as a reliable source of measurement. To examine the relationship between collinearity and orientation, we used a combinatorial algorithm to obtain all possible planes from the given set of points. A statistical criterion of rejecting almost collinear planes was suggested

SPIRE—a software tool for bicontinuous phase recognition: application for plastid cubic membranes

Bicontinuous membranes in cell organelles epitomize nature’s ability to create complex functional nanostructures. Like their synthetic counterparts, these membranes are characterized by continuous membrane sheets draped onto topologically complex saddle-shaped surfaces with a periodic network-like structure. Their structure sizes, (around 50–500 nm), and fluid nature make transmission electron microscopy (TEM) the analysis method of choice to decipher their nanostructural features. Here we present a tool, Surface Projection Image Recognition Environment (SPIRE), to identify bicontinuous structures from TEM sections through interactive identification by comparison to mathematical “nodal surface” models. The prolamellar body (PLB) of plant etioplasts is a bicontinuous membrane structure with a key physiological role in chloroplast biogenesis. However, the determination of its spatial structural features has been held back by the lack of tools enabling the identification and quantitative analysis of symmetric membrane conformations. Using our SPIRE tool, we achieved a robust identification of the bicontinuous diamond surface as the dominant PLB geometry in angiosperm etioplasts in contrast to earlier long-standing assertions in the literature. Our data also provide insights into membrane storage capacities of PLBs with different volume proportions and hint at the limited role of a plastid ribosome localization directly inside the PLB grid for its proper functioning. This represents an important step in understanding their as yet elusive structure–function relationship