An Oracle-based, Output-sensitive Algorithm for Projections of Resultant Polytopes
| Autor: | Emiris, Ioannis Z., Fisikopoulos, Vissarion, Konaxis, Christos, Peñaranda, Luis |
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| Rok vydání: | 2011 |
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| Druh dokumentu: | Working Paper |
| Popis: | We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex- and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is alpha-n-1, where the input consists of alpha points in Z^n. Our approach is output-sensitive as it makes one oracle call per vertex and facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6- and 7-dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in <1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster. Comment: 27 pages, 7 figures, 4 tables. In IJCGA (invited papers from SoCG '12) |
| Databáze: | arXiv |
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