Abstract. Themain contribution ofthiswork is an O(nlogr ~ +k)-timeal gorithmfo rcomputingall k intersections among n line segments in the plane, This time complexity IS easdy shown to be optimal. Within thesame asymptotic cost, ouralgorithm canalso construct thesubdiwslon of theplancdefmed by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.

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This paper describes new methods for maintaining a point-location data structure for a dynamically-changing monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of k-edge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial point-location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point-lo...

We show how to construct an O( p n)-separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)-separator of that subgraph. We also show how to construct an O(n ffl )-way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+ffl )-separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n= log n) processors on a CRCW PRAM. Keywords: Computational geometry, algorithmic graph theory, planar graphs, planar separators, polygon triangulation, parallel algorithms, PRAM model. 1 Introduction Let G = (V; E) be an n-node graph. An f(n)-separator is an f(n)-sized subset of V whose removal disconnects G into two subgraphs G 1 and G 2 each...

In this thesis we study the Input/Output (I/O) complexity of large-scale problems arising e.g. in the areas of database systems, geographic information systems, VLSI design systems and computer graphics, and design I/O-efficient algorithms for them. A general theme in our work is to design I/O-efficient algorithms through the design of I/O-efficient data structures. One of our philosophies is to try to isolate all the I/O specific parts of an algorithm in the data structures, that is, to try to design I/O algorithms from internal memory algorithms by exchanging the data structures used in internal memory with their external memory counterparts. The results in the thesis include a technique for transforming an internal memory tree data structure into an external data structure which can be used in a batched dynamic setting, that is, a setting where we for example do not require that the result of a search operation is returned immediately. Using this technique we develop batched dynamic external versions of the (one-dimensional) range-tree and the segment-tree and we develop an external priority queue. Following our general philosophy we show how these structures can be used in standard internal memory sorting algorithms

We adapt the distribution sweeping method to the cache oblivious model. Distribution sweeping is the name used for a general approach for divide-and-conquer algorithms where the combination of solved subproblems can be viewed as a merging process of streams. We demonstrate by a series of algorithms for specific problems the feasibility of the method in a cache oblivious setting. The problems all come from computational geometry, and are: orthogonal line segment intersection reporting, the all nearest neighbors problem, the 3D maxima problem, computing the measure of a set of axis-parallel rectangles, computing the visibility of a set of line segments from a point, batched orthogonal range queries, and reporting pairwise intersections of axis-parallel rectangles. Our basic building block is a simplified version of the cache oblivious sorting algorithm Funnelsort of Frigo et al., which is of independent interest.

Despite blossoming computer network bandwidths and the emergence of hypertext and CD-ROM databases, little progress has been made towards uniting the world's library-style bibliographic databases. While a few advanced distributed retrieval systems can broadcast a query to hundreds of participating databases, experience shows that local users almost always clog library retrieval systems. Hence broadcast remote queries will clog nearly every system. The premise of this work is that broadcast-based systems do not scale to world-wide systems. This project describes an indexing scheme that will permit thorough yet efficient searches of millions of retrieval systems. Our architecture will work with an arbitrary number of indexing companies and information providers, and, in the market place, could provide economic incentive for cooperation between database and indexing services. We call our scheme distributed indexing, and believe it will help researchers disseminate and locate both publishe...

In the design of algorithms for large-scale applications it is essential to consider the problem of minimizing Input/Output (I/O) communication. Geographical information systems (GIS) are good examples of such large-scale applications as they frequently handle huge amounts of spatial data. In this note we survey the recent developments in external-memory algorithms with applications in GIS. First we discuss the Aggarwal-Vitter I/O-model and illustrate why normal internal-memory algorithms for even very simple problems can perform terribly in an I/O-environment. Then we describe the fundamental paradigms for designing I/O-efficient algorithms by using them to design efficient sorting algorithms. We then go on and survey external-memory algorithms for computational geometry problems -- with special emphasis on problems with applications in GIS -- and techniques for designing such algorithms: Using the orthogonal line segment intersection problem we illustrate the distribution-sweeping and ...

We describe a powerful framework for designing efficient batch algorithms for certain large-scale dynamic problems that must be solved using external memory. The class of problems we consider, which we call colorable external-decomposable problems, include rectangle intersection, orthogonal line segment intersection, range searching, and point location. We are particularly interested in these problems in two and higher dimensions. They have numerous applications in geographic information systems (GIS), spatial databases, and VLSI and CAD design. We present simplified algorithms for problems previously solved by more complicated approaches (such as rectangle intersection), and we present efficient algorithms for problems not previously solved in an efficient way (such as point location and higherdimensional versions of range searching and rectangle intersection). We give experimen...

This paper describes the architecture and the design decisions behind a resource discovery tool that we prototyped to knit together the Internet's resource discovery fabric. We call the architecture distributed indexing or Indie for short. Indie consists of a directory of services and an unlimited number of broker databases that index their own data, data stored in other brokers, and data available from other resource discovery services. The indexing mechanism doubles as a lazily consistent data replication mechanism that can replicate the directory of services or any other broker at will. An Indie broker automatically clusters references to related objects stored in other autonomous discovery and database services. Since Indie brokers cluster related information skimmed from thousands of scattered services, efficient exhaustive search is possible. This centralization led to the success of the archie file location service. In a way, Indie is a generalized archie that locates autonomou...