Efficient geographic information systems: Data structures, Boolean operations and concurrency control

Geographic Information Systems (GIS) are crucial to the ability of govern mental agencies and business to record, manage and analyze geographic data efficiently. They provide methods of analysis and simulation on geographic data that were previously infeasible using traditional hardcopy maps. Creation of realistic 3-D sceneries by overlaying satellite imagery over digital elevation models (DEM) was not possible using paper maps. Determination of suitable areas for construction that would have the fewest environmental impacts once required manual tracing of different map sets on mylar sheets; now it can be done in real time by GIS. Geographic information processing has significant space and time require ments. This thesis concentrates on techniques which can make existing GIS more efficient by considering these issues: Data Structure, Boolean Operations on Geographic Data, Concurrency Control. Geographic data span multiple dimensions and consist of geometric shapes such as points, lines, and areas, which cannot be efficiently handled using a traditional one-dimensional data structure. We therefore first survey spatial data structures for geographic data and then show how a spatial data structure called an R-tree can be used to augment the performance of many existing GIS. Boolean operations on geographic data are fundamental to the spatial anal ysis common in geographic data processing. They allow the user to analyze geographic data by using operators such as AND, OR, NOT on geographic ob jects. An example of a boolean operation query would be, Find all regions that have low elevation AND soil type clay. Boolean operations require signif icant time to process. We present a generalized solution that could significantly improve the time performance of evaluating complex boolean operation queries. Concurrency control on spatial data structures for geographic data processing is becoming more critical as the size and resolution of geographic databases increase. We present algorithms to enable concurrent access to R-tree spatial data structures so that efficient sharing of geographic data can occur in a multi user GIS environment

Single-crossing orthogonal axial lines in orthogonal rectangles

The axial map of a town is one of the key components of the space syntax method – a tool for analysing urban layout. It is derived by placing the longest and fewest lines, called axial lines, to cross the adjacencies between convex polygons in a convex map of a town. Previous research has shown that placing axial lines to cross the adjacencies between a collection of convex polygons is NP-complete, even when the convex polygons are restricted to rectangles and the axial lines to have orthogonal orientation. In this document, we show that placing orthogonal axial lines in orthogonal rectangles where the adjacencies between the rectangles are restricted to be crossed only once (ALPSC- OLOR) is NP-complete. As a result, we infer the single adjacency crossing version of the general axial line placement problem is NP-complete. The transformation of NPcompleteness of ALP-SC-OLOR is from vertex cover for biconnected planar graphs. A heuristic is then presented that gives a reasonable approximate solution to ALP-SC-OLOR based on a greedy method

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input

We study the problem of computing the upper bound of the discrete Fr\'{e}chet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms.Comment: 15 pages, 8 figure

Domain Ordering and Box Cover Problems for Beyond Worst-Case Join Processing

Join queries are a fundamental computational task in relational database management systems. For decades, complex joins were most often computed by decomposing the query into a query plan made of a sequence of binary joins. However, for cyclic queries, this type of query plan is sub-optimal. The worst-case run time of any such query plan exceeds the number of output tuples for any query instance. Recent theoretical developments in join query processing have led to join algorithms which are worst-case optimal, meaning that they run in time proportional to the worst-case output size for any query with the same shape and the same number of input tuples. Building on these results are a class of algorithms providing bounds which go beyond this worst-case output size by exploiting the structure of the input instance rather than just the query shape. One such algorithm, Tetris, is worst-case optimal and also provides an upper bound on its run time which depends on the minimum size of a geometric box certificate for the input query. A box certificate is a subset of a box cover whose union covers every tuple which is not present in the query output. A box cover is a set of n-dimensional boxes which cover all of the tuples not contained in the input relations. Many query instances admit different box certificates and box covers when the values in the attributes' domains are ordered differently. If we permute the input query according to a domain ordering which admits a smaller box certificate, use the permuted query as input to Tetris, then transform the result back with the inverse domain ordering, we can compute the query faster than was possible if the domain ordering was fixed. If we can efficiently compute an optimal domain ordering for a query, then we can state a beyond worst-case bound that is stronger than what is provided by Tetris. This paper defines several optimization problems over the space of domain orderings where the objective is to minimize the size of either the minimum box certificate or the minimum box cover for the given input query. We show that most of these problems are NP-hard. We also provide approximation algorithms for several of these problems. The most general version of the box cover minimization problem we will study, BoxMinPDomF, is shown to be NP-hard, but we can compute an approximation only a poly-logarithmic factor larger than K^(a*r), where K is the minimum box cover size under any domain ordering and r is the maximum number of attributes in a relation. This result allows us to compute join queries in time N+K^(a*r*(w+1))+Z, times a poly-logarithmic factor in N, where N is the number of input tuples, w is the treewidth of the query, and Z is the number of output tuples. This is a new beyond worst-case bound. There are queries for which this bound is exponentially smaller than any bound provided by Tetris. The most general version of the box certificate minimization problem we study, CertMinPDomF, is also shown to be NP-hard. It can be computed exactly if the minimum box certificate size is at most 3, but no approximation algorithm for an arbitrary minimum size is known. Finding such an approximation algorithm is an important direction for future research