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.

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.

This thesis focuses on range searching data structures, an elementary problem in computational geometry with research spanning decades. These problems often involve very large data sets. Processor speeds increase faster than memory speeds, thus the gap between the rate at which CPUs can process data and the rate at which it can be retrieved is increasing. To bridge this gap, various levels of cache are used. Since cache misses are costly, algorithms should be cache-friendly. The input-output (I/O) model was the first model for constructing cache-efficient algorithms, focusing on a two-level memory hierarchy. Algorithms for this model require manual tuning to determine optimal values for hardware dependent parameters, and are only optimal at a single level of a memory hierarchy. Cache-oblivious (CO) algorithms are built without knowledge of the hierarchy, allowing them to be optimal across all levels at once. There exist strong theoretical and practical results for I/O-efficient range searching. Recently, the CO model has received attention, but range searching remains poorly understood. This thesis explores data structures for CO range counting and reporting. It presents the first space and worst-case query-time optimal approximate range counting structure for a family of related problems, and associated O(N log N)-space query-optimal reporting structures. The approximate counting structure is the first of its kind in internal memory, I/O and CO models. Researchers have been trying to create linear-space query-optimal CO reporting structures. This thesis shows that for a variety of problems, linear space is in fact impossible. Heuristics are also used for building cache-friendly algorithms. Space-filling curves are continuous functions mapping multi-dimensional sets into one-dimensional ones. They are used to build search structures in the hopes that objects that were close in the original space remain close in the resulting ordering. This results in queries incurring fewer page swaps when traversing the structure. The Hilbert curve is notably good at this, but often imposes a space or time penalty. This thesis introduces compact Hilbert indices, which remove the ineffiency inherent for input point sets with bounding boxes smaller than their bounding hypercubes.

We consider variations of range searching in which, given a query range, our goal is to compute some function based on frequencies of points that lie in the range. The most basic such computation involves counting the number of points in a query range. Data structures that compute this function solve the well-studied range counting problem. We consider adaptive and approximate data structures for the 2-D orthogonal range counting problem under the w-bit word RAM model. The query time of an adaptive range counting data structure is sensitive to k, the number of points being counted. We give an adaptive data structure that requires O(n loglog n) space and O(loglog n + log_w k) query time. Non-adaptive data structures on the other hand require Ω(log_w n) query time (Pătraşcu, 2007). Our specific bounds are interesting for two reasons. First, when k=O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem (Chan et al., 2011). Second, when k=Θ(n), our data structure is tight to the aforementioned Ω(log_w n) query time lower bound. We also give approximate data structures for 2-D orthogonal range counting whose bounds match the state of the art for the 2-D orthogonal range emptiness problem. Our first data structure requires O(n loglog n) space and O(loglog n) query time. Our second data structure requires O(n) space and O(log^ε n) query time for any fixed constant ε>0. These data structures compute an approximation k' such that (1-δ)k≤k'≤(1+δ)k for any fixed constant δ>0. The range selection query problem in an array involves finding the kth lowest element in a given subarray. Range selection in an array is very closely related to 3-sided 2-D orthogonal range counting. An extension of our technique for 3-sided 2-D range counting yields an efficient solution to adaptive range selection in an array. In particular, we present an adaptive data structure that requires O(n) space and O(log_w k) query time, exactly matching a recent lower bound (Jørgensen and Larsen, 2011). We next consider a variety of frequency-based range query problems in arrays. We give efficient data structures for the range mode and least frequent element query problems and also exhibit the hardness of these problems by reducing Boolean matrix multiplication to the construction and use of a range mode or least frequent element data structure. We also give data structures for the range α-majority and α-minority query problems. An α-majority is an element whose frequency in a subarray is greater than an α fraction of the size of the subarray; any other element is an α-minority. Surprisingly, geometric insights prove to be useful even in the design of our 1-D range α-majority and α-minority data structures.

We consider variations of range searching in which, given a query range, our goal is to compute some function based on frequencies of points that lie in the range. The most basic such computation involves counting the number of points in a query range. Data structures that compute this function solve the well-studied range counting problem. We consider adaptive and approximate data structures for the 2-D orthogonal range counting problem under the w-bit word RAM model. The query time of an adaptive range counting data structure is sensitive to k, the number of points being counted. We give an adaptive data structure that requires O(n loglog n) space and O(loglog n + log_w k) query time. Non-adaptive data structures on the other hand require Ω(log_w n) query time (Pătraşcu, 2007). Our specific bounds are interesting for two reasons. First, when k=O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem (Chan et al., 2011). Second, when k=Θ(n), our data structure is tight to the aforementioned Ω(log_w n) query time lower bound. We also give approximate data structures for 2-D orthogonal range counting whose bounds match the state of the art for the 2-D orthogonal range emptiness problem. Our first data structure requires O(n loglog n) space and O(loglog n) query time. Our second data structure requires O(n) space and O(log^ε n) query time for any fixed constant ε>0. These data structures compute an approximation k' such that (1-δ)k≤k'≤(1+δ)k for any fixed constant δ>0. The range selection query problem in an array involves finding the kth lowest element in a given subarray. Range selection in an array is very closely related to 3-sided 2-D orthogonal range counting. An extension of our technique for 3-sided 2-D range counting yields an efficient solution to adaptive range selection in an array. In particular, we present an adaptive data structure that requires O(n) space and O(log_w k) query time, exactly matching a recent lower bound (Jørgensen and Larsen, 2011). We next consider a variety of frequency-based range query problems in arrays. We give efficient data structures for the range mode and least frequent element query problems and also exhibit the hardness of these problems by reducing Boolean matrix multiplication to the construction and use of a range mode or least frequent element data structure. We also give data structures for the range α-majority and α-minority query problems. An α-majority is an element whose frequency in a subarray is greater than an α fraction of the size of the subarray; any other element is an α-minority. Surprisingly, geometric insights prove to be useful even in the design of our 1-D range α-majority and α-minority data structures.

H διατριβή διερευνά προβλήματα αποδοτικής οργάνωσης χωροταξικών δεδομένων, προτείνει συγκεκριμένες δενδρικές δομές για τη διαχείρισή τους και, τέλος, δίνει παραδείγματα χρήσης τους σε ειδικές περιοχές εφαρμογών. Το πρώτο κεφάλαιο ασχολείται με το γεωμετρικό πρόβλημα της εύρεσης των ισo-προσανατολισμένων ορθογωνίων που περικλείουν ένα query αντικείμενο που μπορεί να είναι ένα ισο-προσανατολισμένο ορθογώνιο είτε σημείο ή κάθετο / οριζόντιο ευθύγραμμο τμήμα. Για την επίλυσή του προτείνεται μια πολυεπίπεδη δενδρική δομή που βελτιώνει τις πολυπλοκότητες των προηγούμενων καλύτερων λύσεων. Το δεύτερο κεφάλαιο εξετάζει το πρόβλημα της ανάκτησης σημείων σε πολύγωνα. H προτεινόμενη γεωμετρική δομή είναι επίσης πολυεπίπεδη και αποδοτική όταν το query πολύγωνο έχει συγκεκριμένες ιδιότητες. Το τρίτο κεφάλαιο ασχολείται με την εφαρμογή δενδρικών δομών σε δύο βιομηχανικά προβλήματα. Το πρώτο αφορά στη μείωση της πολυπλοκότητας ανίχνευσης συγκρούσεων κατά την κίνηση ενός ρομποτικού βραχίονα σε μια επίπεδη σκηνή με εμπόδια. Ο αλγόριθμος επίλυσης κάνει χρήση μιας ουράς προτεραιότητας και μιας UNION-FIND δομής ενώ αξιοποιεί γνωστές δομές και αλγόριθμους της Υπολογιστικής Γεωμετρίας όπως υπολογισμός κυρτών καλυμμάτων, έλεγχος polygon inclusion, κλπ. Το δεύτερο πρόβλημα ασχολείται με το σχεδιασμό απαιτήσεων υλικών (MRP) σε ένα βιομηχανικό σύστημα παραγωγής. Για το σκοπό αυτό αναπτύχθηκε ένας MRP επεξεργαστής που χρησιμοποιεί διασυνδεμένες λίστες και εκτελείται στην κύρια μνήμη για να είναι αποδοτικός. Το τελευταίο κεφάλαιο εξετάζει το πρόβλημα του ελέγχου της παραγωγής και συγκεκριμένα της δρομολόγησης εργασιών. Στο πλαίσιο αυτό σχεδιάστηκε και υλοποιήθηκε ένα ευφυές σύστημα δρομολόγησης σε περιβάλλον ροής που συνδυάζει γνωσιακή τεχνολογία και προσομοίωση με on-line έλεγχο προκειμένου να υποστηρίξει το διευθυντή παραγωγής στη λήψη αποφάσεων. / Τhe dissertation examines problems of efficient organization of spatial data, proposes specific tree structures for their management, and finally, gives examples of their use in specific application areas. The first chapter is about the problem of finding the iso-oriented rectangles that enclose a query object which can be an iso-oriented rectangle either a point or a vertical / horizontal line segment. A multilevel tree structure is proposed to solve the problem which improves the complexities of the best previous known solutions. The second chapter examines the problem of point retrieval on polygons. The proposed geometric structure is also multileveled and efficient when the query polygon has specific properties. The third chapter is about the application of tree structures in two manufacturing problems. The first one concerns the reduction in the complexity of collision detection as a robotic arm moves on a planar scene with obstacles. For the solution a priority queue and a UNION-FIND structure are used, whereas known data structures and algorithms of Computational Geometry such as construction of convex hulls, polygon inclusion testing, etc. are applied. The second problem is about material requirements planning (MRP) in a manufacturing production system. To this end an MRP processor was developed, which uses linked lists and runs in main memory to retain efficiency. The last chapter examines the production control problem, and more specifically the job scheduling problem. In this context, an intelligent scheduling system was designed and developed for flow shop production control which combines knowledge-based technology and simulation with on-line control in order to support the production manager in decision making.