A Framework for Dynamic Terrain with Application in Off-road Ground Vehicle Simulations

The dissertation develops a framework for the visualization of dynamic terrains for use in interactive real-time 3D systems. Terrain visualization techniques may be classified as either static or dynamic. Static terrain solutions simulate rigid surface types exclusively; whereas dynamic solutions can also represent non-rigid surfaces. Systems that employ a static terrain approach lack realism due to their rigid nature. Disregarding the accurate representation of terrain surface interaction is rationalized because of the inherent difficulties associated with providing runtime dynamism. Nonetheless, dynamic terrain systems are a more correct solution because they allow the terrain database to be modified at run-time for the purpose of deforming the surface. Many established techniques in terrain visualization rely on invalid assumptions and weak computational models that hinder the use of dynamic terrain. Moreover, many existing techniques do not exploit the capabilities offered by current computer hardware. In this research, we present a component framework for terrain visualization that is useful in research, entertainment, and simulation systems. In addition, we present a novel method for deforming the terrain that can be used in real-time, interactive systems. The development of a component framework unifies disparate works under a single architecture. The high-level nature of the framework makes it flexible and adaptable for developing a variety of systems, independent of the static or dynamic nature of the solution. Currently, there are only a handful of documented deformation techniques and, in particular, none make explicit use of graphics hardware. The approach developed by this research offloads extra work to the graphics processing unit; in an effort to alleviate the overhead associated with deforming the terrain. Off-road ground vehicle simulation is used as an application domain to demonstrate the practical nature of the framework and the deformation technique. In order to realistically simulate terrain surface interactivity with the vehicle, the solution balances visual fidelity and speed. Accurately depicting terrain surface interactivity in off-road ground vehicle simulations improves visual realism; thereby, increasing the significance and worth of the application. Systems in academia, government, and commercial institutes can make use of the research findings to achieve the real-time display of interactive terrain surfaces

A multifacility location problem on median spaces

AbstractThis paper is concerned with the problem of locating n new facilities in the median space when there are k facilities already located. The objective is to minimize the weighted sum of distances. Necessary and sufficient conditions are established. Based on these results a polynomial algorithm is presented. The algorithm requires the solution of a sequence of minimum-cut problems. The complexity of this algorithm for median graphs and networks and for finite median spaces with ¦V¦points is O(¦V¦3 + ¦V¦ψ(n)), where ψ(n) is the complexity of the applied maximum-flow algorithm. For a simple rectilinear polygon P with N edges and equipped with the rectilinear distance the analogical algorithm requires O(N + k(logN + logk + ψ(n))) time and O(N + kψ(n)) time in the case of the vertex-restricted multifacility location problem

Fully Retroactive Approximate Range and Nearest Neighbor Searching

We describe fully retroactive dynamic data structures for approximate range reporting and approximate nearest neighbor reporting. We show how to maintain, for any positive constant $d$, a set of $n$ points in $\R^d$ indexed by time such that we can perform insertions or deletions at any point in the timeline in $O(\log n)$ amortized time. We support, for any small constant $\epsilon>0$, $(1+\epsilon)$-approximate range reporting queries at any point in the timeline in $O(\log n + k)$ time, where $k$ is the output size. We also show how to answer $(1+\epsilon)$-approximate nearest neighbor queries for any point in the past or present in $O(\log n)$ time.Comment: 24 pages, 4 figures. To appear at the 22nd International Symposium on Algorithms and Computation (ISAAC 2011

Discrete Geometry

[no abstract available

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Algorithmic Motion Planning and Related Geometric Problems on Parallel Machines (Dissertation Proposal)

The problem of algorithmic motion planning is one that has received considerable attention in recent years. The automatic planning of motion for a mobile object moving amongst obstacles is a fundamentally important problem with numerous applications in computer graphics and robotics. Numerous approximate techniques (AI-based, heuristics-based, potential field methods, for example) for motion planning have long been in existence, and have resulted in the design of experimental systems that work reasonably well under various special conditions [7, 29, 30]. Our interest in this problem, however, is in the use of algorithmic techniques for motion planning, with provable worst case performance guarantees. The study of algorithmic motion planning has been spurred by recent research that has established the mathematical depth of motion planning. Classical geometry, algebra, algebraic geometry and combinatorics are some of the fields of mathematics that have been used to prove various results that have provided better insight into the issues involved in motion planning [49]. In particular, the design and analysis of geometric algorithms has proved to be very useful for numerous important special cases. In the remainder of this proposal we will substitute the more precise term of algorithmic motion planning by just motion planning

Permutation classes

This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics