Convex Hull Formation for Programmable Matter

We envision programmable matter as a system of nano-scale agents (called particles) with very limited computational capabilities that move and compute collectively to achieve a desired goal. We use the geometric amoebot model as our computational framework, which assumes particles move on the triangular lattice. Motivated by the problem of sealing an object using minimal resources, we show how a particle system can self-organize to form an object's convex hull. We give a distributed, local algorithm for convex hull formation and prove that it runs in $\mathcal{O}(B)$ asynchronous rounds, where $B$ is the length of the object's boundary. Within the same asymptotic runtime, this algorithm can be extended to also form the object's (weak) $\mathcal{O}$-hull, which uses the same number of particles but minimizes the area enclosed by the hull. Our algorithms are the first to compute convex hulls with distributed entities that have strictly local sensing, constant-size memory, and no shared sense of orientation or coordinates. Ours is also the first distributed approach to computing restricted-orientation convex hulls. This approach involves coordinating particles as distributed memory; thus, as a supporting but independent result, we present and analyze an algorithm for organizing particles with constant-size memory as distributed binary counters that efficiently support increments, decrements, and zero-tests --- even as the particles move

Fundamental Computational Geometry on the GPU

Ph.DDOCTOR OF PHILOSOPH

Algorithm Libraries for Multi-Core Processors

By providing parallelized versions of established algorithm libraries, we ease the exploitation of the multiple cores on modern processors for the programmer. The Multi-Core STL provides basic algorithms for internal memory, while the parallelized STXXL enables multi-core acceleration for algorithms on large data sets stored on disk. Some parallelized geometric algorithms are introduced into CGAL. Further, we design and implement sorting algorithms for huge data in distributed external memory

The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.Comment: This is a major revision; re-organization and better presentation of some parts; correction of several inaccuracies; improvement of some proofs and figures; added references; modification of the title; the paper is long but more than half of it is composed of proofs and references: it is sufficient to look at pages 5, 7--11 in order to understand the algorith

Summary of Research 1994

The views expressed in this report are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.This report contains 359 summaries of research projects which were carried out under funding of the Naval Postgraduate School Research Program. A list of recent publications is also included which consists of conference presentations and publications, books, contributions to books, published journal papers, and technical reports. The research was conducted in the areas of Aeronautics and Astronautics, Computer Science, Electrical and Computer Engineering, Mathematics, Mechanical Engineering, Meteorology, National Security Affairs, Oceanography, Operations Research, Physics, and Systems Management. This also includes research by the Command, Control and Communications (C3) Academic Group, Electronic Warfare Academic Group, Space Systems Academic Group, and the Undersea Warfare Academic Group