Post-Doctoral Research Visit F/M Multipoint evaluation and spatial subdivisions

il y a 1 mois

VillerslèsNancy, France INRIA Temps plein

Contexte et atouts du poste

The INRIA team Gamble is part of the INRIA Nancy Grand Est research center and also of the department Algorithms, Computation, Geometry and Image of the LORIA laboratory of Université de Lorraine.

Classical computational geometry usually deals with linear objects in a Euclidean setting and when other situations happen, curved objects are typically linearized and non-Euclidean spaces are locally approximated by Euclidean spaces. The goals of the Gamble team are to address such limitations of classical computational geometry.

Mission confiée

The goal of this postdoc is to study questions that are relevant both to discrete and computational geometry and to computer algebra, by combining combining classical methods and recent developments from both fields.

As an example, consider the typical problem, underlying several questions in computational geometry and computer algebra, is semi-algebraic range counting: given a set P of n points and a set F of polynomials, count the number of pair (f,p) ∈ F × P such that f(p) >  0. Consider for instance the case where the points are in ℝ2 and the polynomials are bivariate. On one hand, recent results in computational geometry show that if F consists of n polynomials of constant degrees, then it is possible to do it in O(n4/3) operations [1]. On the other hand, results in computer algebra based on a completely different set of tools show that if F is reduced to a single polynomial of degree n1/2, then the problem can be solved in O(n1.334) operations [2]. Is it possible to combine these two sets of tools to obtain better bounds? Or could we obtain lower bounds for those problems, in the spirit of the recent lower bounds obtained for selialgebraic range reporting problems [3]?

State of the art

The main tools from computational geometry that we will consider are cuttings [4, Ch. 4 and 6] and their algebraic generalization [5]. Initially, cuttings were introduced to allow divide-and-conquer algorithms in geometry. For example, given n lines in the plane and a parameter 0 

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