Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

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

Piecewise-Linear Farthest-Site Voronoi Diagrams

Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it. We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2)

Solving k-SUM Using Few Linear Queries

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c<d where d is the ceiling of k/2. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n^3 log^2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in ~O(n^{d+8}) time, and performs O(n^3 log^2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n^3 log^2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-a-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth ~O(n^3) for the k-SUM problem

Non-Crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers

Centerpoint theorems for wedges

Genera

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