Subquadratic Encodings for Point Configurations

For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension

Encoding 3SUM

We consider the following problem: given three sets of real numbers, output a word-RAM data structure from which we can efficiently recover the sign of the sum of any triple of numbers, one in each set. This is similar to a previous work by some of the authors to encode the order type of a finite set of points. While this previous work showed that it was possible to achieve slightly subquadratic space and logarithmic query time, we show here that for the simpler 3SUM problem, one can achieve an encoding that takes $\tilde{O}(N^{\frac 32})$ space for inputs sets of size $N$ and allows constant time queries in the word-RAM

On Order Types of Random Point Sets

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently the space of order types. We present two results on this question. First, we study experimentally the bias in the order types of $n$ random points chosen uniformly and independently in a square, for $n$ up to $16$. Second, we study algorithms for determining the order type of a point set in terms of the number of coordinate bits they require to know. We give an algorithm that requires on average $4n \log\_2 n+O(n)$ bits to determine the order type of $P$, and show that any algorithm requires at least $4n \log\_2 n - O(n \log\log n)$ bits. This implies that the concentration conjecture cannot be proven by an "efficient encoding" argument

Subquadratic Encodings for Point Configurations

For most algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of some realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 (loglog n)^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/loglog n) query time without blowing up the space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.info:eu-repo/semantics/publishe

Algorithms and Data Structures for 3SUM and Friends

Les problèmes 3SUM, k-SUM, et GPT sont considérés comme des problèmes clés de la classe de complexité P en ce qu'ils capturent la complexité de nombreux autres problèmes de cette classe.Malgré leur ancienneté dans le paysage de la géométrie algorithmique, de nombreuses questions au sujet de ces problèmes clés restent encore non résolues.Dans cette thèse nous developpons de nouveaux algorithmes et structures dedonnées pour ces problèmes: A) Nous donnons le premier algorithme efficace pour k-SUM utilisant peu de requêtes pour résoudre le problème. B) Nous définissons un problème intermediaire à 3SUM et GPT, 3POL, et montrons que les techniques existantes pour 3SUM peuvent être généralisées pour obtenir des algorithmes sous-quadratiques pour 3POL. C) Nous montrons que l'information combinatoire contenue dans une instance du problème GPT peut être encodée en un nombre sous-quadratique de bits tout en permettant l'accès efficace à cette information. D) Nous montrons que le nombre de bits d'un tel encodage peut être encore réduit significativement pour les instances obtenues à partir de la réduction du problème 3SUM.Ces nouveaux résultats nous permettent de mieux comprendre la nature fondamentale de ces problèmes. Par exemple, le point B) nous permet de résoudre certaines instances de GPT en temps sous-quadratique en exploitant leur structure.Malgré nos efforts, certaines questions restent encore ouvertes. On ne sait par exemple toujours pas si GPT admet un algorithme en temps sous-quadratique, uniforme ou non, dans les modèles de calculs habituellement étudiés.Nous espérons que les résultats développés dans cette thèse amènerons un jour, directement ou indirectement, à la résolution d'une ou plusieurs de ces questions ouvertes.This thesis is a compilation of the contributions from four papers: A) Solving k-SUM Using Few Linear Queries with Jean Cardinal and John Iacono. B) Subquadratic Algorithms for Algebraic 3SUM with Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, and Noam Solomon. C) Subquadratic Encodings for Point Configurations with Jean Cardinal, Timothy Chan, John Iacono, and Stefan Langerman. D) Encoding 3SUM with Sergio Cabello, Jean Cardinal, John Iacono, Stefan Langerman, and Pat Morin.Option Informatique du Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe