Computational Geometry for Imprecise Data

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Introduction

Computational geometry deals with algorithms on geometric inputs. Historically, geometric algorithms have focused on the ideal situation in which an algorithm runs on a well-defined input and performs its internal calculations using perfect precision. Note the two simplifying assumptions in the previous statement: well-defined input and perfect precision.

Luckily many geometric algorithms rely only on simple geometric predicates that can be evaluated by determining the sign of a low-degree polynomial so that the perfect precision assumed above can be simulated by a fixed precision given as a function of the precision of the input. In other cases, this is not so easy, but techniques have been developed to allow robust geometric computation for a number of problems and scenarios. These techniques are quite mature and even appear in computational geometry libraries as ready-to-use solutions.

The assumption of well-defined-input is the topic of the current workshop. In many cases, geometric data comes from measurements of continuous real-world phenomenon. In almost all cases, the measuring devices used are subject to finite precision. In such cases, the input data given to a geometric algorithm is only an approximation of the real-world phenomenon. Applying a geometric algorithm to this data can often be misleading since the output of such algorithms is usually a combinatorial object such as a convex hull or Delaunay triangulation. However, the structure of this object may be deeply affected by the error in the input, and the actual structure may be very different from that returned by the algorithm.

One solution to this problem is to design algorithms that explicitly compute with imprecise data. This imprecise data can be modelled in different ways. One possible model, for data that consists of points, is to model each point as a region. This region represents all possible locations for where the point might be. Given a collection $S$ of such ``imprecise points'' one can then ask questions about these points. What is their convex hull? What is their Voronoi diagram/Delaunay triangulation?

Of course, the exact structure of the convex hull/Voronoi diagram/Delaunay triangulation depends on exactly where the points are. However, it is possible to summarize the possible convex hulls/Voronoi diagrams/Delaunay triangulations. For convex hulls, one could classify each point as either definitely on the convex hull, definitely not on the convex hull or uncertain. Similarly, a triple of points could be definitely Delaunay, definitely not-Delaunay, or possibly Delaunay. Alternatively, if one is concerned with a worst-case outcome, it may make sense to consider optimization problems on imprecise data. An example of such a question is: What is the smallest (largest) area that the convex hull of $S$ might have.

A related notion is that of robustness. When considering robustness, the set $S$ is a set of observed points. However, it is understood that the observations that make up $S$ are subject to some (known or unknown) level of accuracy. From $S$ one computes a geometric structure such as a convex hull as well as tolerances for (all or part of) this structure. Such tolerances are usually expressed as a value $\epsilon$ whose meaning is ``If no point of $S$ is more than $\epsilon$ from its true location then this (part of the) structure is correct.''

The remainder of this document contains a brief list of some of the papers on the topic of computational geometry for imprecise data and some papers to do with robustness and tolerance. Along with each paper is a brief summary and/or copy of the paper's abstract.

Contents

• Introduction
• Convex Hulls, Delaunay Triangulations, and Voronoi Diagrams
• Range Queries
• Robustness
• Other

Convex Hulls, Delaunay Triangulations, and Voronoi Diagrams

Almost-Delaunay Simplices: Nearest Neighbor Relations for Imprecise Points
Deepak Bandyopadhyay and Jack Snoeyink
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Summary:The authors define almost delaunay simplices, which are sets of points that could form delaunay simplices if all data points were perturbed by at most $\epsilon$. They show that, for a given $k$-simplex $Q$ in a set $S$ of points, finding the smallest value of $\epsilon$ that makes $Q$ almost-delaunay is is equivalent to finding the minimum-width annulus that contains $Q$ and whose interior sphere is empty of points in $S$. They then propose and test algorithms based on this observation.

Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
M. Löffler and J. Snoeyink
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Summary:We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.

Computing Delaunay Triangulation with Imprecise Input Data
A. A. Khanban and A. Edalat
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Summary:The key step in the construction of the Delaunay triangulation of a finite set of planar points is to establish correctly whether a given point of this set is inside or outside the circle determined by any other three points. We address the problem of formulating the in-circle test when the coordinates of the planar points are given only up to a given precision, which is usually the case in practice. By modelling imprecise points as rectangles, and using the idea of partial disc, we construct a reliable in-circle test that provides the best possible Delaunay triangulation with the imprecise input data given by rectangles.

We present a simple algorithm, which is $O(N^4)$, but illustrates the crucial role of the so-called in-circle test in computing the Delaunay Triangulation.

Guaranteed Voronoi Diagrams of Uncertain Sites
W. Evans and J. Sember
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Summary:We investigate the Voronoi diagram that is induced by a set of sites in the plane, where each site's precise location is uncertain but is known to be within a particular region, and the cells of this diagram contain those points guaranteed to be closest to a particular site. We then examine the diagram for sites with disc-shaped regions of uncertainty, prove that it has linear complexity, and provide an optimal O(n log n) algorithm for its construction. We also examine the diagram for polygonal regions of uncertainty, and prove that it has linear complexity as well. We then describe a generalization of these diagrams, in which each Voronoi cell is associated with a subset of the sites, and each point in a cell is guaranteed to be closest to some site in the subset associated with the cell.

Largest and Smallest Convex Hulls for Imprecise Points
Maarten Löffler and Marc van Kreveld
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Summary:Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from $O(n\log n)$ to $O(n^13 )$, and prove NP-hardness for some other variants.

Approximating Largest Convex Hulls for Imprecise Points
Maarten Löffler and Marc van Kreveld
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Summary:Assume that a set of imprecise points is given, where each point is specified by a region in which the point will lie. Such a region can be modelled as a circle, square, line segment, etc. We study the problem of maximising the area of the convex hull of such a set. We prove NP-hardness when the imprecise points are modelled as line segments, and give linear time approximation schemes for a variety of models, based on the core-set paradigm.

Computational Geometry with Imprecise Input Data
A. Edalat, A. Khanban, and A. Lieutier
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Summary:The authors consider an imprecise input data model where each input ``point'' is represented as a convex polygon that includes the set of all possible locations for that point. The output of an algorithm is a pair of open sets that the interior and exterior of a partially defined geometric object. In this model, the authors consider Voronoi diagrams, Delaunay triangulations, and convex hulls. In the case where the polygons have $O(1)$ sides, these algorithms have $O(n\log n)$ running times(?)

Range Queries

Efficient Indexing Methods for Probabilistic Threshold Queries over Uncertain Data
Reynold Cheng, Yuni Xia, Sunil Prabhakar, Rahul Shah, Jeffrey Scott Vitter
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Summary:In this applied paper, the authors consider the following scenario. A set of data points $x_1,\ldots,x_n$ exist, where each $x_i$ is a random variable whose probability distribution is known. This data is then processed so that it is possible to answer any probabilistic threshold query $(a,b,p)$ of the form: Report all $x_i$ whose values are in the interval $[a,b]$ with probability at least $p$. The authors show how to answer these queries using a variant of R-trees and give experimental results. They also prove that this problem is at least as hard as preprocessing for simplex range queries in 2D and halfspace range queries in 3D.

Querying Imprecise Data in Moving Object Environments
R. Cheng, D. V. Kalashnikov and S. Prabhakar
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Summary:In this paper we study the execution of probabilistic range and nearest-neighbor queries. The imprecision in answers to queries is an inherent property of these applications due to uncertainty in data, unlike the techniques for approximate nearest-neighbor processing that trade accuracy for performance. The notion of probabilistic answers to queries over uncertain data was introduced by Wolfson et al. for range queries, where the answer consists of objects along with the probability of each object lying in the query range. We extend this idea to answer nearest-neighbor queries – the answer consists of not a single object that is closest to the object, but a set of objects each of which have the potential of being the nearest neighbor of the query point. In addition to identifying these objects, the probability of each object being the nearest neighbor can also be evaluated.

Engineering approach, no theoretical bounds.

Efficient Evaluation of Imprecise Location-Dependent Queries
J. Chen and R. Cheng
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Summary:Studies the imprecise version of range queries. The query returns the identities of all objects that fall within the region. We propose efficient algorithms to compute qualification probabilities. Moreover, we develop a set of novel filtering techniques to determine quickly the region which contains objects that may satisfy the query. In particular, we classify a query according to whether the location data being accessed is (1) precise (e.g., locations of gas stations, schools etc.), or (2) uncertain (e.g., locations of moving objects).

Engineering approach, no theoretical bounds Also provide an I/O efficient version.

Robustness

Epsilon Geometry: Building Robust Algorithms from Imprecise Computations
L. Guibas, D. Salesin and J. Stolfi
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Summary:We describe a new general framework, called Epsilon Geometry, for coping with computational errors in geometric algorithms that arise from the use of finite precision arithmetic. The Epsilon Geometry framework allows us to build robust geometric algorithms out of imprecise geometric primitives. Our method combines the techniques of interval arithmetic and backward error analysis, along with a good deal of geometric reasoning. Our algorithms compute an exact solution for a perturbed version of the input, and they return a bound on the size of this implicit perturbation.

Other

Largest Bounding Box, Smallest Diameter, and Related Problems on Imprecise Points
Maarten Löffler and Marc van Kreveld
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Summary:We model imprecise points as regions in which one point must be located. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding box. We give efficient algorithms for most of these problems, and identify the hardness of others.

Existence of Simple Tours of Imprecise Points
Maarten Löffler
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Summary:Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NP-complete to decide whether it is possible to place the points inside their regions in such a way that the resulting polygon is simple. Furthermore, it is NP-hard to minimize the length of a simple tour visiting the regions in order, when the connections between consecutive regions do not need to be straight line segments.

Agent Performance in Vehicle Routing when the Only Thing Certain is Uncertainty
Jordan Srour, Tamas Mahr, Rob Zuidwijk, and Mathijs de Weerdt
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Summary:The authors consider a vehicle routing problem in which some fraction of the requests are known in advance and some fraction arrive online. The propose an agent-based solution for this problem and give computational results for their solution.

This idea might lead to an interesting extension of online algorithm and the competitive ratio to semi-online algorithms. For several problems (paging, list organization, etc.) the (randomized and deterministic) competitive ratio of these algorithms is fully-understood. What effect does giving some of the access sequence in advance have on the competitive ratio for these problems?

Computation with Imprecise Geospatial Data
Michael Worboys
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Summary:Attempts to formalize imprecision in the context of GIS.

Basic Algorithms of Computational Geometry with Imprecise Input (PhD Thesis)
Ali Asghar Khanban
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Summary:This PhD thesis contains the results of some of Khanban's other papers listed here.

Regular Polygons are Most Tolerant
William Evans
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Summary:The author shows that a regular $n$-gon is the most tolerant in the sense that, among all convex $n$-gons, the regular $n$-gon requires the largest perturbation of its vertices to make it non-convex. This answers a question posed by Abellanas, Hurtado, and Ramos (1994).

Geometry with Imprecise Lines
M. Löffler, M. van Kreveld
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Summary:Proposes definitions for convex sets of lines(?)

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Computational Geometry for Imprecise Data

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