Fast Detection of Degenerate Predicates in Free Space Construction

Victor Milenkovic, Elisha Sacks, Nabeel Butt

Research output: Contribution to journalArticle

Abstract

An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial f evaluated at a zero t of a univariate polynomial g, where the coefficients of f and g are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

Original languageEnglish (US)
Pages (from-to)219-237
Number of pages19
JournalInternational Journal of Computational Geometry and Applications
Volume29
Issue number3
DOIs
StatePublished - Sep 1 2019

Fingerprint

Free Space
Predicate
Polynomials
Polyhedron
Univariate
Polynomial
Degeneracy
Zero
Computational geometry
Highest common factor
Common factor
Multivariate Polynomials
Computational Geometry
Factoring
Robustness
Ring
Output
Coefficient
Vertex of a graph

Keywords

  • configuration spaces
  • multivariate polynomial factoring
  • Robust computational geometry

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Cite this

Fast Detection of Degenerate Predicates in Free Space Construction. / Milenkovic, Victor; Sacks, Elisha; Butt, Nabeel.

In: International Journal of Computational Geometry and Applications, Vol. 29, No. 3, 01.09.2019, p. 219-237.

Research output: Contribution to journalArticle

@article{b63f056ec100456399ca455c089bbc3b,
title = "Fast Detection of Degenerate Predicates in Free Space Construction",
abstract = "An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial f evaluated at a zero t of a univariate polynomial g, where the coefficients of f and g are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90{\%} to 0.5{\%} of the running time.",
keywords = "configuration spaces, multivariate polynomial factoring, Robust computational geometry",
author = "Victor Milenkovic and Elisha Sacks and Nabeel Butt",
year = "2019",
month = "9",
day = "1",
doi = "10.1142/S0218195919500067",
language = "English (US)",
volume = "29",
pages = "219--237",
journal = "International Journal of Computational Geometry and Applications",
issn = "0218-1959",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "3",

}

TY - JOUR

T1 - Fast Detection of Degenerate Predicates in Free Space Construction

AU - Milenkovic, Victor

AU - Sacks, Elisha

AU - Butt, Nabeel

PY - 2019/9/1

Y1 - 2019/9/1

N2 - An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial f evaluated at a zero t of a univariate polynomial g, where the coefficients of f and g are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

AB - An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial f evaluated at a zero t of a univariate polynomial g, where the coefficients of f and g are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

KW - configuration spaces

KW - multivariate polynomial factoring

KW - Robust computational geometry

UR - http://www.scopus.com/inward/record.url?scp=85074360864&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074360864&partnerID=8YFLogxK

U2 - 10.1142/S0218195919500067

DO - 10.1142/S0218195919500067

M3 - Article

AN - SCOPUS:85074360864

VL - 29

SP - 219

EP - 237

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 3

ER -