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

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Contract type : Fixed-term contract

Level of qualifications required : PhD or equivalent

Fonction : Post-Doctoral Research Visit

Context

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.

Assignment

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 < ε < 1, a ε−cutting is a triangulation of ℝ2 such that each triangle contains at most εn lines.

In computer algebra, the main tools used for fast bivariate polynomial evaluation are based on fast matrix multiplication [6, Ch. 12]. This tool was used to compute the modular composition of polynomials [7], and later extended to evaluate a bivariate polynomial on multiplie points [2].

An important data structure underlying these approaches is the biclique partition. Given a subset S of a product of sets F × P, a biclique partition of S is a union of pairwise distinct subsets Si ⊂ S of the form Fi × Pi ⊂ F × P such that S = ∪iSi. This kind of data structure appears notably in several geometric incidence and algebraic rigidity problems. Such problems include bounding the number of line intersections, or bounding the number of intersection of an algebraic surface with a Cartesian product [8].

[1] P. K. Agarwal, B. Aronov, E. Ezra, M. J. Katz, and M. Sharir, “Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems,” in 38th international symposium on computational geometry (SoCG 2022), 2022, vol. 224, pp. 4:1–4:14, doi: 10.4230/LIPIcs.SoCG.2022.4.
[2] M. Nüsken and M. Ziegler, “Fast multipoint evaluation of bivariate polynomials,” in Algorithms – ESA 2004. 12th annual european symposium, bergen, norway, september 14–17, 2004. proceedings., Berlin: Springer, 2004, pp. 544–555.
[3] P. Afshani and P. Cheng, “On semialgebraic range reporting,” Discrete Comput. Geom., vol. 71, no. 1, pp. 4–39, 2024, doi: 10.1007/s00454-023-00574-1.
[4] J. Matoušek, Lectures on discrete geometry, vol. 212. New York, NY: Springer, 2002.
[5] H. Kaplan, J. Matoušek, and M. Sharir, “Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique,” Discrete Comput. Geom., vol. 48, no. 3, pp. 499–517, 2012, doi: 10.1007/s00454-012-9443-3.
[6] J. von zur Gathen and J. Gerhard, Modern computer algebra, 3rd ed. Cambridge: Cambridge University Press, 2013.
[7] R. P. Brent and H. T. Kung, “Fast algorithms for manipulating formal power series,” J. Assoc. Comput. Mach., vol. 25, pp. 581–595, 1978, doi: 10.1145/322092.322099.
[8] G. Elekes and E. Szabó, “How to find groups?” Combinatorica, vol. 32, no. 5, pp. 537–571, 2012, doi: 10.1007/s00493-012-2505-6.

Main activities

The candidate will study the recent results on semi-algebraic-relational queries to try and narrow the gaps between upper and lower bounds. They will try to improve asymptotic bounds on related problems in computational geometry and computer algebra, and to compare and combine the methods from both fields.

Depending on the interests of the candidate, the postdoc can open up in a number of directions. For example, these problems are naturally studied in the Real RAM model of computation, but a comparison to the word-RAM model is relevant (in particular, it would be interesting to inspect the frontier between the classes NP and ETR, if any). Other possiblities include, but are not limited to, investigating from a computer algebra standpoint problems in dicrete and computational geometry where polynomials play a key role (e.g. the various phenomena of algebraic rigidity that arise in incidence geometry, semi-algebraic graphs arising from geometry, certain LP-type optimization problems, etc.).

Skills

Academic background: Ph.D. in computer science or mathematics.

Required knowledge and skills: algorithms in discrete and computational geometry.

Benefits package

  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • Access to vocational training
  • Social security coverage

Remuneration

2788 € gross/month