Expansive Motions and the Polytope of Pointed Pseudo-Triangulations

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1-dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially to the "Further remarks" in Section 5) + changed to final book format. This version is to appear in "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), series "Algorithms and Combinatorics", Springer Verlag, Berli

Minimal knotted polygons in cubic lattices

An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data are collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe

Practical methods for approximating shortest paths on a convex polytope in R3

AbstractWe propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p, q on its surface, let dP(p, q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2dP(p, q) in time O(n). Extending this result, we can also compute an approximation of the shortest path tree rooted at an arbitrary point x ∈ P in time O(n log n). In the approximate tree, the distance between a vertex v ∈ P and x is at most cdP(x, v), where c = 2.38(1 + ε) for any fixed ε > 0. The best algorithms for computing an exact shortest path on a convex polytope take Ω(n2) time in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in O(n) time our algorithm gives a path of length at most 2k times the optimal

Ground-Penetrating-Radar Reflection Attenuation Tomography with an Adaptive Mesh

Ground-penetrating radar (GPR) attenuation-difference analysis can be a useful tool for studying fluid transport in the subsurface. Surface-based reflection attenuation-difference tomography poses a number of challenges that are not faced by crosshole attenuation surveys. We create and analyze a synthetic attenuation-difference GPR data set to determine methods for processing amplitude changes and inverting for conductivity differences from reflection data sets. Instead of using a traditional grid-based inversion, we use a data-driven adaptive-meshing algorithm to alter the model space and to create amore even distribution of resolution. Adaptive meshing provides a method for improving the resolution of the model space while honoring the data limitations and improving the quality of the attenuation difference inversion. Comparing inversions on a conventional rectangular grid with the adaptive mesh, we find that the adaptively meshed model reduces the inversion computation time by an average of 75% with an improvement in the root mean square error of up to 15%. While the sign of the conductivity change is correctly reproduced by the inversion algorithm, the magnitude varies by as much as much as 50% from the true values. Our heterogeneous conductivity model indicates that the attenuation difference inversion algorithm effectively locates conductivity changes, and that surface-based reflection surveys can produce models as accurate as traditional crosshole surveys

Geodesics on an ellipsoid of revolution

Algorithms for the computation of the forward and inverse geodesic problems for an ellipsoid of revolution are derived. These are accurate to better than 15 nm when applied to the terrestrial ellipsoids. The solutions of other problems involving geodesics (triangulation, projections, maritime boundaries, and polygonal areas) are investigated.Comment: LaTex, 29 pages, 16 figures. Supplementary material is available at http://geographiclib.sourceforge.net/geod.htm

Travelling Salesman Problem with Neighborhoods

Práce se zabývá využitím metaheuristického algoritmu GLNS, používaného k řešení problému obecného obchodního cestujícího, k řešení upraveného problému obchodního cestujícího se sousedstvími. Tato úprava spočívá v tom, že sousedstvími jsou pouze nedegenerované mnohoúhelníky, jež se mohou i překrývat. V rámci práce jsou navrhnuty a implementovány dva algorithmy, které využívají původní nebo modifikovaný algoritmus GLNS. Dále je v obou také využit algoritmus pro řešení úlohy průchodu mnohoúhelníky. Druhý navrhnutý algoritmus je schopný řešit i instance, kde jsou mezi sousedstvími překážky ve tvaru nedegenerovaných mnohoúhelníků. Využívá k tomu datové struktury, která se nazývá graf viditelnosti.This thesis explores the possibility of transforming the metaheuristic algorithm GLNS, used for General Travelling Salesman Problem (GTSP), to instead solve the version of Travelling Salesman Problem with Neighborhoods (TSPN) where the neighborhoods are simple, possibly intersecting, polygons. Two algorithms are proposed and implemented, each utilizing GLNS in a different way. Both also make use of an algorithm solving the unconstrained version of Touring Polygons Problem (TPP). The second proposed algorithm is additionally equipped to handle a case, when there are simple polygonal obstacles between the neighborhoods. This is made possible using a visibility graph