Fast algorithms for collision and proximity problems involving moving geometric objects

Consider a set of geometric objects, such as points, line segments, or axes-parallel hyperrectangles in \IR^d, that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search

On Geometric Range Searching, Approximate Counting and Depth Problems

In this thesis we deal with problems connected to range searching, which is one of the central areas of computational geometry. The dominant problems in this area are halfspace range searching, simplex range searching and orthogonal range searching and research into these problems has spanned decades. For many range searching problems, the best possible data structures cannot offer fast (i.e., polylogarithmic) query times if we limit ourselves to near linear storage. Even worse, it is conjectured (and proved in some cases) that only very small improvements to these might be possible. This inefficiency has encouraged many researchers to seek alternatives through approximations. In this thesis we continue this line of research and focus on relative approximation of range counting problems. One important problem where it is possible to achieve significant speedup through approximation is halfspace range counting in 3D. Here we continue the previous research done and obtain the first optimal data structure for approximate halfspace range counting in 3D. Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast to the previous methods that were Monte Carlo (the correctness holds with high probability). Another series of problems where approximation can provide us with substantial speedup comes from robust statistics. We recognize three problems here: approximate Tukey depth, regression depth and simplicial depth queries. In 2D, we obtain an optimal data structure capable of approximating the regression depth of a query hyperplane. We also offer a linear space data structure which can answer approximate Tukey depth queries efficiently in 3D. These data structures are obtained by applying our ideas for the approximate halfspace counting problem. Approximating the simplicial depth turns out to be much more difficult, however. Computing the simplicial depth of a given point is more computationally challenging than most other definitions of data depth. In 2D we obtain the first data structure which uses near linear space and can answer approximate simplicial depth queries in polylogarithmic time. As applications of this result, we provide two non-trivial methods to approximate the simplicial depth of a given point in higher dimension. Along the way, we establish a tight combinatorial relationship between the Tukey depth of any given point and its simplicial depth. Another problem investigated in this thesis is the dominance reporting problem, an important special case of orthogonal range reporting. In three dimensions, we solve this problem in the pointer machine model and the external memory model by offering the first optimal data structures in these models of computation. Also, in the RAM model and for points from an integer grid we reduce the space complexity of the fastest known data structure to optimal. Using known techniques in the literature, we can use our results to obtain solutions for the orthogonal range searching problem as well. The query complexity offered by our orthogonal range reporting data structures match the most efficient query complexities known in the literature but our space bounds are lower than the previous methods in the external memory model and RAM model where the input is a subset of an integer grid. The results also yield improved orthogonal range searching in higher dimensions (which shows the significance of the dominance reporting problem). Intersection searching is a generalization of range searching where we deal with more complicated geometric objects instead of points. We investigate the rectilinear disjoint polygon counting problem which is a specialized intersection counting problem. We provide a linear-size data structure capable of counting the number of disjoint rectilinear polygons intersecting any rectilinear polygon of constant size. The query time (as well as some other properties of our data structure) resembles the classical simplex range searching data structures

Algorithms for learning from spatial and mobility data

Data from the numerous mobile devices, location-based applications, and collection sensors used currently can provide important insights about human and natural processes. These insights can inform decision making in designing and optimising in frastructure such as transportation or energy. However, extracting patterns related to spatial properties is challenging due to the large quantity of the data produced and the complexity of the processes it describes. We propose scalable, multi-resolution approximation and heuristic algorithms that make use of spatial proximity properties to solve fundamental data mining and optimisation problems with a better running time and accuracy. We observe that abstracting from individual data points and working with units of neighbouring points based on various measures on similarity, improves computational efficiency and diminishes the effects of noise and overfitting. We consider applications in: mobility data compression, transit network planning, and solar power output prediction. Firstly, in order to understand transportation needs, it is essential to have efficient ways to represent large amounts of travel data. In analysing spatial trajectories (for example taxis travelling in a city), one of the main challenges is computing distances between trajectories efficiently; due to their size and complexity this task is computationally expensive. We build data structures and algorithms to sketch trajectory data that make queries such as distance computation, nearest neighbour search and clustering, which are key to finding mobility patterns, more computationally efficient. We use locality sensitive hashing, a technique that associates similar objects to the same hash. Secondly, to build efficient infrastructure it is necessary to satisfy travel demand by placing resources optimally. This is difficult due to external constraints (such as limits on budget) and the complexity of existing road networks that allow for a large number of candidate locations. For this purpose, we present heuristic algorithms for efficient transit network design with a case study on cycling lane placement. The heuristic is based on a new type of clustering by projection, that is both computationally efficient and gives good results in practice. Lastly, we devise a novel method to forecast solar power output based on numerical weather predictions, clear sky predictions and persistence data. The ensemble of a multivariate linear regression model, support vector machines model, and an artificial neural network gives more accurate predictions than any of the individual models. Analysing the performance of the models in a suite of frameworks reveals that building separate models for each self-contained area based on weather patterns gives a better accuracy than a single model that predicts the total. The ensemble can be further improved by giving performance-based weights to the individual models. This suggests that the models identify different patterns in the data, which motivated the choice of an ensemble architecture