809 research outputs found

    Improved time-space trade-offs for computing Voronoi diagrams

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    Let P be a planar set of n sites in general position. For k∈{1,
,n−1}, the Voronoi diagram of order k for P is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest k neighbors in P. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of k=1 and k=n−1, respectively. For any given K∈{1,
,n−1}, the family of all higher-order Voronoi diagrams of order k=1,
,K for P can be computed in total time O(nK2+nlogn) using O(K2(n−K)) space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for P can be computed in O(nlogn) time using O(n) space [Preparata, Shamos, Springer'85]. For s∈{1,
,n} , an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words, of Θ(logn) bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards. We describe a deterministic s -workspace algorithm for computing NVD and FVD for P that runs in O((n2/s)logs) time. Moreover, we generalize our s-workspace algorithm so that for any given K∈O(s√), we compute the family of all higher-order Voronoi diagrams of order k=1,
,K for P in total expected time O(n2K5s(logs+K2O(log∗K))) or in total deterministic time O(n2K5s(logs+KlogK)). Previously, for Voronoi diagrams, the only known s-workspace algorithm runs in expected time O((n2/s)logs+nlogslog∗s) [Korman et al., WADS'15] and only works for NVD (i.e., k=1). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques

    Improved Time-Space Trade-Offs for Computing Voronoi Diagrams

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    Let P be a planar n-point set in general position. For k between 1 and n-1, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1, respectively. It is known that the family of all higher-order Voronoi diagrams of order 1 to K for P can be computed in total time O(n K^2 + n log n) using O(K^2(n-K)) space. Also NVD and FVD can be computed in O(n log n) time using O(n) space. For s in {1, ..., n}, an s-workspace algorithm has random access to a read-only array with the sites of P in arbitrary order. Additionally, the algorithm may use O(s) words of Theta(log n) bits each for reading and writing intermediate data. The output can be written only once and cannot be accessed afterwards. We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our s-workspace algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K in O(sqrt(s)) in total time O( (n^2 K^6 / s) log^(1+epsilon)(K) (log s / log K)^(O(1)) ) for any fixed epsilon > 0. Previously, for Voronoi diagrams, the only known s-workspace algorithm was to find an NVD for P in expected time O((n^2/s) log s + n log s log^*s). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques

    Fast and Compact Exact Distance Oracle for Planar Graphs

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    For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an O(n5/3)O(n^{5/3})-space distance oracle which answers exact distance queries in O(log⁥n)O(\log n) time for nn-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space O(n2−ϔ)O(n^{2 - \epsilon}) for some constant Ï”>0\epsilon > 0) either required query time polynomial in nn or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any S≄n3/2S \ge n^{3/2}, we show how to obtain an SS-space distance oracle that answers queries in time O((n5/2/S3/2)log⁥n)O((n^{5/2}/ S^{3/2}) \log n). This is a polynomial improvement over the previous planar distance oracles with o(n1/4)o(n^{1/4}) query time

    A Time-Space Tradeoff for Triangulations of Points in the Plane

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    In this paper, we consider time-space trade-offs for reporting a triangulation of points in the plane. The goal is to minimize the amount of working space while keeping the total running time small. We present the first multi-pass algorithm on the problem that returns the edges of a triangulation with their adjacency information. This even improves the previously best known random-access algorithm

    Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees

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    In the limited-workspace model, we assume that the input of size nn lies in a random access read-only memory. The output has to be reported sequentially, and it cannot be accessed or modified. In addition, there is a read-write workspace of O(s)O(s) words, where s∈{1,
,n}s \in \{1, \dots, n\} is a given parameter. In a time-space trade-off, we are interested in how the running time of an algorithm improves as ss varies from 11 to nn. We present a time-space trade-off for computing the Euclidean minimum spanning tree (EMST) of a set VV of nn sites in the plane. We present an algorithm that computes EMST(V)(V) using O(n3log⁡s/s2)O(n^3\log s /s^2) time and O(s)O(s) words of workspace. Our algorithm uses the fact that EMST(V)(V) is a subgraph of the bounded-degree relative neighborhood graph of VV, and applies Kruskal's MST algorithm on it. To achieve this with limited workspace, we introduce a compact representation of planar graphs, called an ss-net which allows us to manipulate its component structure during the execution of the algorithm

    The Limited Workspace Model for Geometric Algorithms

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    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 AnfĂ€ngen des Algorithmenentwurfs von Interesse. Die erhöhte VerfĂŒgbarkeit von GerĂ€ten 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 fĂŒr dieses Thema gefĂŒhrt. In der Folge hat der Entwurf von Algorithmen fĂŒr das Limited Workspace Model (Modell mit begrenztem Arbeitsspeicher) in den letzten zehn Jahren einen signifikanten Anstieg an PopularitĂ€t in der algorithmischen Geometrie erfahren. In diesem Setting haben wir in der Regel eine große Menge an Daten, die verarbeitet werden mĂŒssen. 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, ĂŒbergeordneten Speicher (z. B. CPU-Register) abgelegt. Da die Anwendungsbereiche der oben genannten GerĂ€te – 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 GrĂ¶ĂŸe n in einem schreibgeschĂŒtzten Speicher, ein Algorithmus kann einen Arbeitsspeicher der GrĂ¶ĂŸe s = {1, . . . , n} verwenden, um die Zwischendaten wĂ€hrend der AusfĂŒhrung zu lesen und zu schreiben. Die Ausgabe sendet er an einen lesegeschĂŒtzten Stream. Ziel ist es, Algorithmen zu entwickeln, deren Laufzeit mit zunehmender VerfĂŒgbarkeit an Arbeitsspeicher abnimmt, was einen Time-Space Trade-Off (Laufzeit-Speicher-AbwĂ€gung) darstellt. In dieser Arbeit betrachten wir drei grundlegende geometrische Probleme, nĂ€mlich 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 fĂŒr diese Probleme oder verbessern die bisherigen Ergebnisse
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