Problems In Computational Geometry Research Articles

Enabling high-dimensional range queries using kNN indexing techniques: approaches and empirical results

Many modern search applications are high-dimensional and depend on efficient orthogonal range queries. These applications span web-based and scientific needs as well as uses for data mining. Although k-nearest neighbor queries are becoming increasingly common due to mobile and geospatial applications, orthogonal range queries in high-dimensional data remain extremely important and relevant. For efficient querying, data is typically stored in an index optimized for either kNN or range queries. This can be problematic when data is optimized for kNN retrieval and a user needs a range query or vice versa. Here, we address the issue of using a kNN-based index for range queries, as well as outline the general computational geometry problem of adapting these systems to range queries. We refer to these methods as space-based decompositions and provide a straightforward heuristic for this problem. Using iDistance as our applied kNN indexing technique, we also develop an optimal (data-based) algorithm designed specifically for its indexing scheme. We compare this method to the suggested naive approach using real world datasets. The data-based algorithm consistently performs better.

Targeting Aggregate Production Planning for an Energy Supply Chain

Aggregate production planning for an energy supply chain is presented in this paper. The methodology is equally applicable to aggregate production planning of production as well as aggregate planning of input material supply. The proposed methodology determines the range of production at different times as well as calculates the production targets with least variation in production levels between different time periods to satisfy the demands of each time period while maintaining the limits of the inventory. Mathematically, it is shown that the graphical representation of the aggregate production planning problem is equivalent to the Euclidean shortest path problem in computational geometry. An algebraic procedure, based on the principles of Pinch Analysis, is applied to solve the problem. The proposed graphical representation provides significant physical understanding of the overall aggregate production planning problem by identifying different kinds of production bottlenecks (also known as pinch points)...

Fast Wavefront Propagation (FWP) for Computing Exact Geodesic Distances on Meshes.

Computing geodesic distances on triangle meshes is a fundamental problem in computational geometry and computer graphics. To date, two notable classes of algorithms, the Mitchell-Mount-Papadimitriou (MMP) algorithm and the Chen-Han (CH) algorithm, have been proposed. Although these algorithms can compute exact geodesic distances if numerical computation is exact, they are computationally expensive, which diminishes their usefulness for large-scale models and/or time-critical applications. In this paper, we propose the fast wavefront propagation (FWP) framework for improving the performance of both the MMP and CH algorithms. Unlike the original algorithms that propagate only a single window (a data structure locally encodes geodesic information) at each iteration, our method organizes windows with a bucket data structure so that it can process a large number of windows simultaneously without compromising wavefront quality. Thanks to its macro nature, the FWP method is less sensitive to mesh triangulation than the MMP and CH algorithms. We evaluate our FWP-based MMP and CH algorithms on a wide range of large-scale real-world models. Computational results show that our method can improve the speed by a factor of 3-10.

Comparison of three approaches to studying stability of solutions to problems of discrete optimization and computational geometry

In the 1970–1980s, V. K. Leont’ev and E. N. Gordeev proposed and studied an approach to the analysis of the stability of solutions. In a number of subsequent articles, this approach was developed and the stability of solutions was analyzed on its basis. The approach itself is of a rather general nature but was originally connected with the discrete optimization problems. Later similar results, although in different terms, were published for various classes of problems. We demonstrate the proximity of approaches both on the level of statements of problems and the interpretation of results.

Swath-acquisition planning in multiple-satellite missions: an exact and heuristic approach

This paper deals with the swath acquisition planning problem for multisatellite Earth observation missions. Given a set of satellites and a mission time frame, the problem we solve consists of selecting a set of acquisitions from the satellites in order to cover a given region of interest during the requested time frame, optimizing a certain objective function.We show that the planning problem can be modeled as a set covering problem, using basic tools of mathematical programming. The formulation of the model requires the solution of a complex computational geometry problem, and therefore the use of heuristics and metaheuristics applies. In this paper, we discuss the efficiency of the constructive phase of a greedy randomized adaptive search procedure algorithm. Computational results comparing the heuristic algorithms with the exact approach are presented.

Four-Connected Triangulations of Planar Point Sets

In this paper, we consider the problem of determining in polynomial time whether a given planar point set $$P$$P of $$n$$n points in general position admits a 4-connected triangulation. We propose a necessary and sufficient condition for recognizing such point sets $$P$$P, and present an $$O(n^3)$$O(n3) time algorithm for constructing a 4-connected triangulation of $$P$$P, if it exists. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple $$O(n^2)$$O(n2) time method for constructing a non-complex triangulation of $$P$$P, if it exists. This method provides a new insight into the structure of 4-connected triangulations of point sets.

Computing Shortest Paths among Curved Obstacles in the Plane

A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this article, we consider the problem version on curved obstacles, which are commonly modeled as splinegons . A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons), and the combinatorial complexity of each curved edge is assumed to be O (1). Given in the plane two points s and t and a set s of h pairwise disjoint splinegons with a total of n vertices, after a bounded degree decomposition of S is obtained, we compute a shortest s -to- t path avoiding the splinegons in O ( n + h log h + k ) time, where k is a parameter sensitive to the geometric structures of the input and is upper bounded by O ( h 2 ). The bounded degree decomposition of S , which is similar to the triangulation of the polygonal domains, can be computed in O ( n log n ) time or O ( n + h log 1 + ϵ h ) time for any ϵ > 0. In particular, when all splinegons are convex, the decomposition can be computed in O ( n + h log h ) time and k is linear to the number of common tangents in the free space (called “free common tangents”) among the splinegons. Our techniques also improve several previous results: (1) For the polygon case (i.e., when all splinegons are polygons), the shortest path problem was previously solved in O ( n log n ) time, or in O ( n + h 2 log n ) time. Thus, our algorithm improves the O ( n + h 2 log n ) time result, and is faster than the O ( n log n ) time solution for sufficiently small h , for example, h = o (√ n ,log n . (2) Our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among h pairwise disjoint convex splinegons with a total of n vertices. Our algorithm runs in O ( n + h log h + k ) time and O ( n ) working space, where k is the number of all free common tangents. Note that k = O ( h 2 ). Even for the special case where all splinegons are convex polygons , the previously best algorithm for this visibility problem takes O ( n + h 2 log n ) time. (3) We improve the previous work for computing the shortest path between two points among convex pseudodisks of O (1) complexity each. In addition, a by-product of our techniques is an optimal O ( n + h log h ) time and O ( n ) space algorithm for computing the Voronoi diagram of a set of h pairwise disjoint convex splinegons with a total of n vertices.

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this paper, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, includingan algorithm for the 2D maxima problem that uses $$n\lg h + O(n\sqrt{\lg h})$$nlgh+O(nlgh) comparisons, where $$h$$h denotes the output size; a randomized algorithm for the 3D maxima problem that uses $$n\lg h + O(n\lg ^{2/3} h)$$nlgh+O(nlg2/3h) expected number of comparisons; a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses $$n\lg n + O(n\sqrt{\lg n})$$nlgn+O(nlgn) expected number of comparisons; a data structure for point location among 3D disjoint axis-parallel boxes that can answer queries in $$(3/2)\lg n + O(\lg \lg n)$$(3/2)lgn+O(lglgn) comparisons; a data structure for point location in a 3D box subdivision that can answer queries in $$(4/3)\lg n + O(\sqrt{\lg n})$$(4/3)lgn+O(lgn) comparisons. Some of the results can be adapted to solve nonorthogonal problems, such as 2D convex hulls and general line segment intersection. Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees.

Constrained Reachability and Controllability Sets for Planetary Precision Landing via Convex Optimization

This paper presents a convex optimizations-based method to compute the set of initial conditions from which a given landing accuracy to a target can be achieved (constrained controllability set) and the set of states that can be reached from a given set of initial states (constrained reachability set) for a planetary landing vehicle with all the relevant control and mission constraints. The proposed method is based on the lossless convexification of the powered-descent landing guidance problem and methods of convex optimization and computational geometry. These techniques are used to generate approximations that can be arbitrarily close to the actual reachability or controllability sets. The quantification of these sets allows evaluation of the feasibility of a prescribed landing accuracy for a given vehicle and an expected set of dispersions from the parachute descent phase of a planetary landing mission. Since these sets are generated systematically and quickly, a wide range of design options can be evaluated for different mission requirements. Consequently, the proposed method can enable lander vehicle design optimization as a reliable analysis tool for systematic design and system engineering.

Current Open Problems in Discrete and Computational Geometry

We have selected problems that may not yet be well known, but have the potential to push the research in interesting directions. In particular, we state problems that do not require specific knowledge outside the standard circle of ideas in discrete geometry. Despite the relatively simple statements, these problems are related to current research and their solutions are likely to require new ideas and approaches. We have chosen problems from different fields to make this short paper attractive to a wide range of specialists.The article is published in the author’s wording.

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line: Algorithms and Complexity

Given a set of $$n$$n axis-parallel rectangles in the plane, finding a maximum independent set ($$\mathrm {MIS}$$MIS), a maximum weighted independent set ($$\mathrm {WMIS}$$WMIS), and a minimum hitting set ($$\mathrm {MHS}$$MHS), are basic problems in computational geometry and combinatorics. They have attracted significant attention since the sixties, when Wegner conjectured that the duality gap, equal to the ratio between the size of $$\mathrm {MIS}$$MIS and the size of $$\mathrm {MHS}$$MHS, is always bounded by a universal constant. An interesting case is when there exists a diagonal line that intersects each of the given rectangles. Indeed, Chepoi and Felsner recently gave a 6-approximation algorithm for $$\mathrm {MHS}$$MHS in this setting, and showed that the duality gap is between 3/2 and 6. We consider the same setting and improve upon these results. First, we derive an $$O(n^2)$$O(n2)-time algorithm for the $$\mathrm {WMIS}$$WMIS when, in addition, every pair of intersecting rectangles have a common point below the diagonal. This improves and extends a classic result of Lubiw, and gives a 2-approximation algorithm for $$\mathrm {WMIS}$$WMIS. Second, we show that $$\mathrm {MIS}$$MIS is NP-hard. Finally, we prove that the duality gap is between 2 and 4. The upper bound, which implies a 4-approximation algorithm for $$\mathrm {MHS}$$MHS, follows from simple combinatorial arguments, whereas the lower bound represents the best known lower bound on the duality gap, even in the general setting of the rectangles.

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs $G_1,\dots,G_k$ on the same set $V$ of $n$ vertices, the simultaneous geometric embedding (with mapping) problem, or simply $k$-SGE, is to find a set $P$ of $n$ points in the plane and a bijection $\phi: V \to P$ such that the induced straight-line drawings of $G_1,\dots,G_k$ under $\phi$ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x) proved to be complete for $\exists\mathbb{R}$, the existential theory of the reals. Hence the problem $k$-SGE is polynomial-time equivalent to several other problems in computational geometry, such as recognizing intersection graphs of line segments or finding the rectilinear crossing number of a graph. We give an elementary reduction from the pseudoline stretchability problem to $k$-SGE, with the property that both numbers $k$ and $n$ are linear in the number of pseudolines. This implies not only the $\exists\mathbb{R}$-hardness result, but also a $2^{2^{\Omega (n)}}$ lower bound on the minimum size of a grid on which any such simultaneous embedding can be drawn. This bound is tight. Hence there exists such collections of graphs that can be simultaneously embedded, but every simultaneous drawing requires an exponential number of bits per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is only $2^{2^{\Omega (\sqrt{n})}}$.

Analytical solution from vector potentials for the gravitational field of a general polyhedron

Closed form solutions in terms of elementary functions are given for the Newtonian gravitational field and potential of a general constant density polyhedron, using a gravitational vector potential formulation. The solution constants are given in terms of scalar and vector products involving the position vectors of the field point and the polyhedron’s vertices, giving one analytical expression for each edge of the polyhedron. It is shown that in this vector potential formulation, the gravitational problem is related to the point-in-polygon problem of computational geometry. The solution is derived using a new vector theorem giving the integral over a flat surface of a scalar potential as a line integral of a vector potential around its boundary. The method is also valid in the interior of the polyhedron and can be extended to polyhedra with linear spatial variation of the density. Numerical results are compared with results in the literature and detailed results are given for the five regular Platonic polyhedra.

Formal study of functional orbits in finite domains

In computer science, functional orbits – i.e., tracks left by the iterations of a function – in a finite domain naturally appear at a high or a low level. This paper introduces a Coq logical orbit framework, the purpose of which is to help rigorously developing software systems with some complex data structures from specification to implementation.The result is a Coq library of orbit concepts formalized as definitions, lemmas and theorems. Most of them are inspired by our previous work in geometric modelling, where combinatorial hypermaps were used to describe surface subdivisions appearing in computational geometry problems, e.g., building convex hulls or Delaunay diagrams. Now, this domain remains a reference for us, but our results are drastically extended and usable in other computer science areas. The proof of Floyd's cycle-detection algorithm, known as “the tortoise and the hare”, confirms that point.The library contains operations to observe, traverse and update orbits – addition, deletion, mutation, transposition – with proofs of their behavior. It focuses on the important case where the involved function is a partial injection. In this case, it defines a connectivity relationship and evaluates the variation of the number of connected components during updates.

Symmetry detection in 3D chain coded discrete curves and trees

Symmetry detection in 2D and 3D shapes has been a classical problem in computer vision, computational geometry, and pattern recognition. One of the reasons of this interest, is because symmetry is related to the balance and harmony, and in most cases it is considered like a beautiful proportion. In this paper we present different methods to detect mirror and rotational symmetries in 2D and 3D open and closed curves, and in trees represented by means of the orthogonal direction change chain code. These methods detect both global and local symmetries. They can also detect the number and position of the axes of symmetry. They are simple, easy to implement and fast to execute.

Data imprecision under λ-geometry model

An essential assumption in most algorithms in computational geometry is that the input data is precise which is not always true in real world problems. In this paper we introduce a model – called λ-geometry model – for handling a dynamic form of imprecision which allows the precision changes in the input data of geometric problems. Also, we study two families of basic computational geometry problems under this model: axis-aligned bounding box (AABB) and proximity problems. For a set of n imprecise points where each point is modeled by a convex polygonal region with O(k) representation complexity, we propose an O(n(log n+log k+m)) time algorithm to solve the largest AABB problem, where m is the maximum number of corner defining functions when the largest AABB is defined by points on its corners. Also, we propose an O(n log n+nk+k2 log kα(k)) time algorithm for finding the smallest AABB under the assumption that the regions are disjoint. We show that the algorithms are optimal when k is O(log n). Further, we propose O(n2k log(nk) α(nk)) time algorithms for finding the closest and furthest pair problems, and present a lower bound of Ω(n2) for the closest pair problem when k is constant.

A characterization theorem and an algorithm for a convex hull problem

Given $$S= \{v_1, \dots , v_n\} \subset {\mathbb {R}}^m$$ and $$p \in {\mathbb {R}}^m$$ , testing if $$p \in conv(S)$$ , the convex hull of $$S$$ , is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean distance duality, distinct from classical separation theorems such as Farkas Lemma: $$p$$ lies in $$conv(S)$$ if and only if for each $$p' \in conv(S)$$ there exists a pivot, $$v_j \in S$$ satisfying $$d(p',v_j) \ge d(p,v_j)$$ . Equivalently, $$p \not \in conv(S)$$ if and only if there exists a witness, $$p' \in conv(S)$$ whose Voronoi cell relative to $$p$$ contains $$S$$ . A witness separates $$p$$ from $$conv(S)$$ and approximate $$d(p, conv(S))$$ to within a factor of two. Next, we describe the Triangle Algorithm: given $$\epsilon \in (0,1)$$ , an iterate, $$p' \in conv(S)$$ , and $$v \in S$$ , if $$d(p, p') < \epsilon d(p,v)$$ , it stops. Otherwise, if there exists a pivot $$v_j$$ , it replace $$v$$ with $$v_j$$ and $$p'$$ with the projection of $$p$$ onto the line $$p'v_j$$ . Repeating this process, the algorithm terminates in $$O(mn \min \{ \epsilon ^{-2}, c^{-1}\ln \epsilon ^{-1} \})$$ arithmetic operations, where $$c$$ is the visibility factor, a constant satisfying $$c \ge \epsilon ^2$$ and $$\sin (\angle pp'v_j) \le 1/\sqrt{1+c}$$ , over all iterates $$p'$$ . In particular, the geometry of the input data may result in efficient complexities such as $$O(mn \root t \of {\epsilon ^{-2}} \ln \epsilon ^{-1})$$ , $$t$$ a natural number, or even $$O(mn \ln \epsilon ^{-1})$$ . Additionally, (i) we prove a strict distance duality and a related minimax theorem, resulting in more effective pivots; (ii) describe $$O(mn \ln \epsilon ^{-1})$$ -time algorithms that may compute a witness or a good approximate solution; (iii) prove generalized distance duality and describe a corresponding generalized Triangle Algorithm; (iv) prove a sensitivity theorem to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods. Finally, we discuss future work on applications and generalizations.

Querying visible points in large obstructed space

Querying visible points is a fundamental problem in computational geometry and spatial databases. With the development of new applications such as trip planning and nearest neighbors, querying visible points plays a key role in obstacle space and the result can be further used such as defining the shortest path. Thereby, efficiently finding the result is essentially important. However, the performance of current methods decrease substantially for large datasets. To solve the problem, we proposes a new and fast algorithm to find visible points for an arbitrary query location inside a large polygon containing obstacles. The method is based on polygon triangulation. By decomposing the polygon into a set of triangles, we manage the polygon by organizing triangles in an efficient way instead of maintaining a large number of vertices. We propose a data structure to partition the searching space into several parts, each of which is independently processed. Afterwards, by recursively calling a method we search visible points by accessing triangles and return the result in a progressive way. Through a theoretical analysis, assuming the polygon contains N vertices in total, the time complexity of our algorithm is O(N), improving the existing method O(N l o g N). We prove the correctness of the algorithm and analyze the space complexity, which is O(N). The technique is extended to return visible points less than a threshold distance to the query location. Using both synthetic and real datasets, we perform extensive experiments to test our algorithm and demonstrate its efficiency and effectiveness. Visible points are efficiently processed in a large obstacle space with over one million vertices. Experimental results show that our technique gains more than one order of magnitude speedup compared to competitive methods using large datasets.

Visibility and ray shooting queries in polygonal domains

Given a polygonal domain (or polygon with holes) in the plane, we study the problem of computing the visibility polygon of any query point. As a special case of visibility problems, we also study the ray-shooting problem of finding the first point on the polygon boundaries that is hit by any query ray. These are fundamental problems in computational geometry and have been studied extensively. We present new algorithms and data structures that improve the previous results.

Optimizing the Layout of Proportional Symbol Maps: Polyhedra and Computation

Proportional symbol maps are a cartographic tool to assist in the visualization and analysis of quantitative data associated with specific locations, such as earthquake magnitudes, oil well production, and temperature at weather stations. As the name suggests, symbol sizes are proportional to the magnitude of the physical quantities that they represent. We present two novel integer linear programming (ILP) models to solve this computational geometry problem: how to draw opaque disks on a map so as to maximize the total visible border of all disks. We focus on drawings obtained by layering symbols on top of each other, also known as stacking drawings. We introduce decomposition techniques as well as several families of facet-defining inequalities, which are used to strengthen the ILP models that are supplied to a commercial solver. We demonstrate the effectiveness of our approach through a series of computational experiments using hundreds of instances generated from real demographic and geophysical data sets. To the best of our knowledge, we are the first to use ILP to tackle this problem, and the first to provide provably optimal symbol maps for those data sets.