Space-Time Trade-offs for Stack-Based Algorithms

In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memory-constrained algorithms. Given an algorithm \alg\ that runs in O(n) time using $\Theta(n)$ variables, we can modify it so that it runs in $O(n^2/s)$ time using a workspace of O(s) variables (for any $s\in o(\log n)$) or $O(n\log n/\log p)$ time using $O(p\log n/\log p)$ variables (for any $2\leq p\leq n$). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives the first trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms

Engineering technology for networks

Space Network (SN) modeling and evaluation are presented. The following tasks are included: Network Modeling (developing measures and metrics for SN, modeling of the Network Control Center (NCC), using knowledge acquired from the NCC to model the SNC, and modeling the SN); and Space Network Resource scheduling

An efficient output-sensitive hidden surface removal algorithm and its parallelization

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running time is sensitive to the actual size of the visible image rather than the total number of intersections in the image plane which can be much larger than the visible image. The time complexity of this algorithm is O((k +nflognloglogn) where n and k are respectively the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time Ω(n 2) irrespective of the output size (where as the output size k is O(n 2) only in the worst case). We also present a parallel algorithm based on a similar approach which runs in time O(log4(n+k)) using O((n + k)/Iog(n+k)) processors in a CREW PRAM model. All our bounds arc obtained using ammortized analysis

The Limited Workspace Model for Geometric Algorithms

Space usage has been a concern since the very early days of algorithm design. The increased availability of devices with limited memory or power supply – such as smartphones, drones, or small sensors – as well as the proliferation of new storage media for which write access is comparatively slow and may have negative effects on the lifetime – such as flash drives – have led to renewed interest in the subject. As a result, the design of algorithms for the limited workspace model has seen a significant rise in popularity in computational geometry over the last decade. In this setting, we typically have a large amount of data that needs to be processed. Although we may access the data in any way and as often as we like, write-access to the main storage is limited and/or slow. Thus, we opt to use only higher level memory for intermediate data (e.g., CPU registers). Since the application areas of the devices mentioned above – sensors, smartphones, and drones – often handle a large amount of geographic (i.e., geometric) data, the scenario becomes particularly interesting from the viewpoint of computational geometry. Motivated by these considerations, we investigate geometric problems in the limited workspace model. In this model the input of size n resides in read-only memory, an algorithm may use a workspace of size s = {1, . . . , n} to read and write the intermediate data during its execution, and it reports the output to a write-only stream. The goal is to design algorithms whose running time decreases as s increases, which provides a time-space trade-off. In this thesis, we consider three fundamental geometric problems, namely, computing different types of Voronoi diagrams of a planar point set, computing the Euclidean minimum spanning tree of a planar point set, and computing the k-visibility region of a point inside a polygonal domain. Using several innovative techniques, we either achieve the first time-space trade-offs for those problems or improve the previous results.Der Speicherplatzbedarf ist seit den Anfängen des Algorithmenentwurfs von Interesse. Die erhöhte Verfügbarkeit von Geräten mit begrenztem Speicherplatz oder begrenzter Stromversorgung – wie Smartphones, Drohnen oder kleine Sensoren – sowie die Verbreitung neuer Speichermedien, bei denen der Schreibzugriff vergleichsweise langsam ist und negative Auswirkungen auf die Lebensdauer haben kann – wie beispielsweise Flash-Laufwerken – haben zu erneuter Aufmerksamkeit für dieses Thema geführt. In der Folge hat der Entwurf von Algorithmen für das Limited Workspace Model (Modell mit begrenztem Arbeitsspeicher) in den letzten zehn Jahren einen signifikanten Anstieg an Popularität in der algorithmischen Geometrie erfahren. In diesem Setting haben wir in der Regel eine große Menge an Daten, die verarbeitet werden müssen. Obwohl wir auf die Daten beliebig oft und in beliebiger Weise zugreifen können, ist der Schreibzugriff auf den Hauptspeicher begrenzt und/oder langsam. Zwischenergebnisse werden daher nur in einem kleineren, übergeordneten Speicher (z. B. CPU-Register) abgelegt. Da die Anwendungsbereiche der oben genannten Geräte – Sensoren, Smartphones und Drohnen – oft mit einer großen Menge an geografischen (d. h., geometrischen) Daten umgehen, ist dieses Szenario aus Sicht der algorithmischen Geometrie besonders interessant. Motiviert durch diese Überlegungen haben wir geometrische Probleme im Limited Workspace Model untersucht. In diesem Modell befindet sich die Eingabe der Größe n in einem schreibgeschützten Speicher, ein Algorithmus kann einen Arbeitsspeicher der Größe s = {1, . . . , n} verwenden, um die Zwischendaten während der Ausführung zu lesen und zu schreiben. Die Ausgabe sendet er an einen lesegeschützten Stream. Ziel ist es, Algorithmen zu entwickeln, deren Laufzeit mit zunehmender Verfügbarkeit an Arbeitsspeicher abnimmt, was einen Time-Space Trade-Off (Laufzeit-Speicher-Abwägung) darstellt. In dieser Arbeit betrachten wir drei grundlegende geometrische Probleme, nämlich die Berechnung verschiedener Arten von Voronoi-Diagrammen einer Punktmenge in der Ebene, die Berechnung des euklidischen minimalen Spannbaums einer ebenen Punktmenge und die Bestimmung der k-Sichtbarkeitsregion (k-visibility region) eines Punkts innerhalb eines polygonalen Gebiets. Mit mehreren innovativen Techniken entwickeln wir entweder die ersten Time-Space Trade-Offs für diese Probleme oder verbessern die bisherigen Ergebnisse