Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

On the complexity of range searching among curves

Modern tracking technology has made the collection of large numbers of densely sampled trajectories of moving objects widely available. We consider a fundamental problem encountered when analysing such data: Given $n$ polygonal curves $S$ in $\mathbb{R}^d$, preprocess $S$ into a data structure that answers queries with a query curve $q$ and radius $\rho$ for the curves of $S$ that have \Frechet distance at most $\rho$ to $q$. We initiate a comprehensive analysis of the space/query-time trade-off for this data structuring problem. Our lower bounds imply that any data structure in the pointer model model that achieves $Q(n) + O(k)$ query time, where $k$ is the output size, has to use roughly $\Omega\left((n/Q(n))^2\right)$ space in the worst case, even if queries are mere points (for the discrete \Frechet distance) or line segments (for the continuous \Frechet distance). More importantly, we show that more complex queries and input curves lead to additional logarithmic factors in the lower bound. Roughly speaking, the number of logarithmic factors added is linear in the number of edges added to the query and input curve complexity. This means that the space/query time trade-off worsens by an exponential factor of input and query complexity. This behaviour addresses an open question in the range searching literature: whether it is possible to avoid the additional logarithmic factors in the space and query time of a multilevel partition tree. We answer this question negatively. On the positive side, we show we can build data structures for the \Frechet distance by using semialgebraic range searching. Our solution for the discrete \Frechet distance is in line with the lower bound, as the number of levels in the data structure is $O(t)$, where $t$ denotes the maximal number of vertices of a curve. For the continuous \Frechet distance, the number of levels increases to $O(t^2)$

Encontrando estruturas geométricas com número de trespasse mínimo

Orientador: Cid Carvalho de SouzaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Problemas de trespasse têm sido investigados há tempos em Geometria Computacional pois aplicações para eles são encontradas em uma grande variedade de áreas. Em geral, a entrada é formada por dois conjuntos de objetos geométricos: o conjunto, finito ou infinito, L de trespassadores e o conjunto O. Uma solução viável é um subconjunto O' de O satisfazendo uma certa propriedade estrutural $\pi$. Dado O', o número de trespasse de l em L é a quantidade de elementos de O' intersectados por l. O número de trespasse de O' relativo a L é o número de trespasse máximo dentre qualquer l em L. O objetivo do problema é achar um subconjunto de O satisfazendo a propriedade $\pi$ com o menor número de trespasse possível relativo a L. Esta tese traz contribuições tanto teóricas quanto experimentais para alguns problemas de trespasse. Em [19, 20], Fekete, Lübbecke e Meijer resolveram o problema aberto a respeito da complexidade de encontrar uma árvore geradora com número de trespasse mínimo. Eles também mostraram que achar um emparelhamento perfeito com número de trespasse mínimo é NP-difícil. Modelos de programação inteira para os problemas foram apresentados. Porém, muito poucos experimentos computacionais foram realizados. Nesta tese, estudamos modelos de programação inteira para encontrar emparelhamentos perfeitos, árvores geradoras e triangulação com número de trespasse mínimo. Com base nestas formulações, apresentamos algoritmos exatos e heurísticas Lagrangianas para resolvê-los. Estes algoritmos mostraram que as heurísticas Lagrangianas proveem boas soluções, frequentemente ótimas, em um breve tempo computacional. De todos os dez problemas e variantes discutidos em [20], para apenas três deles a complexidade não foi provada: Triangulação com Número de Trespasse Mínimo, com trespassadores paralelos aos eixos e gerais, e Triangulação com Número de Cruzamento Mínimo, caso geral. Nesta tese, provamos que estes três problemas são NP-difíceis. Outro problema de trespasse mínimo é apresentado em [2] e também estudado em [16]. Este problema pede por uma partição retangular com número de trespasse mínimo em um polígono retilinear. Embora a complexidade do problema ainda seja desconhecida, em [2] um algoritmo de 3-aproximação é apresentado. Em [16] um modelo de programação inteira é dado e uma 2-aproximação reivindicada. Nesta tese, fortalecemos a formulação introduzida em [16]. Também propomos um modelo alternativo e comparamos os dois teórica e computacionalmente. Além disso, mostramos que o algoritmo proposto em [16] não provê uma 2-aproximação para o problemaAbstract: Stabbing problems have long being investigated in Computational Geometry since applications for them are found in a great variety of areas. In general, the input is formed by two sets of geometrical objects: the finite or infinite set L of stabbers and the set O. A feasible solution for the problem is a subset O' of O satisfying a given structural property $\pi$. Given O', the stabbing number of l in L is the total of elements of O' that are intersected by l. The stabbing number of L relative to O' is the maximum stabbing number of all its elements. The goal of the problem is to find a subset of O satisfying property $\pi$ and having the smallest possible stabbing number. This thesis brings both theoretical and experimental contributions to the investigation of some stabbing problems. The works of Fekete, Lübbecke and Meijer [19, 20] solved the open problem relative to the complexity of finding a spanning tree with minimum stabbing number. They also showed that finding a perfect matching with minimum stabbing number is NP-hard. Integer programming formulations for the problems were also presented. However, very few computational experiments were performed. In this thesis we study integer programming formulations for the problems of finding perfect matchings, spanning trees and triangulations with minimum stabbing number. Based on these formulations we present exact algorithms and Lagrangian heuristics to solve the problems. These algorithms show that the Lagrangian heuristics yield solutions with good quality, often optimal, in short time span. Of all the ten problems and variants discussed in [20], for only three of them the complexity was not proved: The Minimum Stabbing Triangulation, axis-parallel and general cases, and The Minimum Crossing Triangulation, general case. In this thesis we prove that the three problems are NP-hard. Another problem of finding a structure with minimum stabbing number is presented in [2] and also studied in [16]. This problem asks for a rectangular partition with minimum stabbing number in a rectilinear polygon. Although the complexity of the problem is still unkown, in [2] a 3-approximation algorithm is presented. In [16] an integer programming formulation is given and a 2-approximation is claimed. In this thesis we strengthen the formulation introduced in [16]. We also propose an alternative model and compare the formulations both theoretically and computationally. Furthermore, we show that the algorithm proposed in [16] can not provide a 2-approximation for the problemDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação147619/2010-6CNP

Stabbing Orthogonal Objects in 3-Space

We consider a problem that arises in the design of data structures for answering {\em visibility range queries}, that is, given a $3$-dimensional scene defined by a set of polygonal patches, we wish to preprocess the scene to answer queries involving the set of patches of the scene that are visible from a given range of points over a given range of viewing directions. These data structures recursively subdivide space into cells until some criterion is satisfied. One of the important problems that arise in the construction of such data structures is that of determining whether a cell represents a nonempty region of space, and more generally computing the size of a cell. In this paper we introduce a measure of the {\em size} of the subset of lines in 3-space that stab a given set of $n$ polygonal patches, based on the maximum angle and distance between any two lines in the set. Although the best known algorithm for computing this size measure runs in $O(n^2)$ time, we show that if the polygonal patches are orthogonal rectangles, then this measure can be approximated to within a constant factor in $O(n)$ time. (Also cross-referenced as UMIACS-TR-96-71

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Algorithms for the Analysis of Spatio-Temporal Data from Team Sports

Modern object tracking systems are able to simultaneously record trajectories—sequences of time-stamped location points—for large numbers of objects with high frequency and accuracy. The availability of trajectory datasets has resulted in a consequent demand for algorithms and tools to extract information from these data. In this thesis, we present several contributions intended to do this, and in particular, to extract information from trajectories tracking football (soccer) players during matches. Football player trajectories have particular properties that both facilitate and present challenges for the algorithmic approaches to information extraction. The key property that we look to exploit is that the movement of the players reveals information about their objectives through cooperative and adversarial coordinated behaviour, and this, in turn, reveals the tactics and strategies employed to achieve the objectives. While the approaches presented here naturally deal with the application-specific properties of football player trajectories, they also apply to other domains where objects are tracked, for example behavioural ecology, traffic and urban planning