Theory and Practice of I/O efficient Algorithms for Multidimensional Batched Searching Problems

Extended AbstractWe 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 higher dimensional versions of range searching and rectangle intersection). We give experimental results concerning the running time for our approach applied to the red-blue rectangle intersection problem, which is a key component of the extremely important database operation spatial join. Our algorithm scales well with the problem size, and for large problems sizes it greatly outperforms the well-known sweepline approach

External-Memory Algorithms for Processing Line Segments in Geographic Information Systems

The original publication is available at www.springerlink.comIn the design of algorithms for large-scale applications it is essential to consider the problem of minimizing 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 paper we develop e cient new external-memory algorithms for a number of important problems involving line segments in the plane, including trapezoid decomposition, batched planar point location, triangulation, red-blue line segment intersection reporting, and general line segment intersection reporting. In GIS systems, the rst three problems are useful for rendering and modeling, and the latter two are frequently used for overlaying maps and extracting information from them

Dynamic Orthogonal Range Searching on the RAM, Revisited

We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n / log log n + k) optimal query time and O(log^{2/3+o(1)}n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen\u27s previous result [SIAM J. Comput., 2006], which has O(log^{7/8+epsilon}n) update time for an arbitrarily small constant epsilon. In the case of 3-sided queries, our update time reduces to O(log^{1/2+epsilon}n), improving Wilkinson\u27s previous bound [ESA 2014] of O(log^{2/3+epsilon}n)

Data structures

We discuss data structures and their methods of analysis. In particular, we treat the unweighted and weighted dictionary problem, self-organizing data structures, persistent data structures, the union-find-split problem, priority queues, the nearest common ancestor problem, the selection and merging problem, and dynamization techniques. The methods of analysis are worst, average and amortized case

Algorithms for Unit-Disk Graphs and Related Problems

In this dissertation, we study algorithms for several problems on unit-disk graphs and related problems. The unit-disk graph can be viewed as an intersection graph of a set of congruent disks. Unit-disk graphs have been extensively studied due to many of their applications, e.g., modeling the topology of wireless sensor networks. Lots of problems on unit-disk graphs have been considered in the literature, such as shortest paths, clique, independent set, distance oracle, diameter, etc. Specifically, we study the following problems in this dissertation: L1 shortest paths in unit-disk graphs, reverse shortest paths in unit-disk graphs, minimum bottleneck moving spanning tree, unit-disk range reporting, distance selection, etc. We develop efficient algorithms for these problems and our results are either first-known solutions or somehow improve the previous work. Given a set P of n points in the plane and a parameter r \u3e 0, a unit-disk graph G(P) can be defined using P as its vertex set and two points of P are connected by an edge if the distance between these two points is at most r. The weight of an edge is one in the unweighted case and is equal to the distance between the two endpoints in the weighted case. Note that the distance between two points can be measured by different metrics, e.g., L1 or L2 metric. In the first problem of L1 shortest paths in unit-disk graphs, we are given a point set P and a source point s ∈ P, the problem is to find all shortest paths from s to all other vertices in the L1 weighted unit-disk graph defined on set P. We present an O(n log n) time algorithm, which matches the Ω(n log n)-time lower bound. In the second problem, we are given a set P of n points, parameters r, λ \u3e 0, and two points s and t of P, the goal is to compute the smallest r such that the shortest path length between s and t in the unit-disk graph with respect to set P and parameter r is at most λ. This problem can be defined in both unweighted and weighted cases. We propose an algorithm of O(⌊λ⌋ · n log n) time and another algorithm of O(n5/4 log7/4 n) time for the unweighted case. We also given an O(n5/4 log5/2 n) time algorithm for the weighted case. In the third problem, we are given a set P of n points that are moving in the plane, the problem is to compute a spanning tree for these moving points that does not change its combinatorial structure during the point movement such that the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement is minimized. We present an algorithm that runs in O(n4/3 log3 n) time. The fourth problem is unit-disk range reporting in which we are given a set P of n points in the plane and a value r, we need to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We build a data structure of O(n) space in O(n log n) time that can answer each query in O(k + log n) time, where k is the output size. The time complexity of our algorithm is the same as the previous result but our approach is much simpler. Finally, for the problem of distance selection, we are given a set P of n points in the plane and an integer 1 ≤ k ≤ (n2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of p. We propose an algorithm that runs in O(n4/3 log n) time. Our techniques yield two algorithmic frameworks for solving geometric optimization problems. Many algorithms and techniques developed in this dissertation are quite general and fundamental, and we believe they will find other applications in future