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An Output-Sensitive Algorithm for Computing Projections of Resultant Polytopes

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An Output-Sensitive Algorithm for Computing Projections of Resultant Polytopes An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes ∗ ∗ Ioannis Z. Emiris Vissarion Fisikopoulos Department of Informatics & Telecommunications Department of Informatics & Telecommunications University of Athens, Athens, Greece University of Athens, Athens, Greece [email protected] [email protected] ∗∗ ∗ Christos Konaxis Luis Peñaranda Archimedes Center for Modeling, Department of Informatics & Telecommunications Analysis & Computation (ACMAC) University of Athens, Athens, Greece University of Crete, Heraklio, Greece [email protected] [email protected] ABSTRACT Categories and Subject Descriptors We develop an incremental algorithm to compute the New- F.2.2 [Analysis of Algorithms and Problem Complex- ton polytope of the resultant, aka resultant polytope, or its ity]: Nonnumerical Algorithms and Problems|Geometri- projection along a given direction. The resultant is fun- cal problems and computations; G.4 [Mathematical Soft- damental in algebraic elimination and in implicitization of ware]: Algorithm design and analysis. parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired poly- General Terms tope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per Algorithms, Design, Experimentation, Theory. vertex. We overcome the bottleneck of determinantal predi- cates by hashing, thus accelerating execution from 18 to 100 Keywords times. We implement our algorithm using the experimental General Dimension, Convex Hull, Regular Triangulation, CGAL package triangulation. A variant of the algorithm Secondary Polytope, Resultant, CGAL Implementation, Ex- computes successively tighter inner and outer approxima- perimental Complexity. tions: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. 1. INTRODUCTION Our method computes instances of 5-, 6- or 7-dimensional n polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Given pointsets A0;:::;An ⊂ Z , we define the pointset Compared to tropical geometry software, ours is faster up n [ 2n to dimension 5 or 6, and competitive in higher dimensions. A := (Ai × feig) ⊂ Z ; (1) i=0 n ∗This research was partially funded by the University of where e0; : : : ; en form an affine basis of R : e0 is the zero vec- o Athens' Special Account of Research Grants n 10812. Par- tor, ei = (0;:::; 0; 1; 0;:::; 0); i = 1; : : : ; n. Clearly, jAj = tial support from project \Computational Geometric Learn- jA0j + ··· + jAnj, where j · j denotes cardinality. By Cayley's ing", which acknowledges the financial support of the Fu- trick (Prop. 2) the regular tight mixed subdivisions of the ture and Emerging Technologies (FET) programme within Minkowski sum A0+···+An are in bijection with the regular the 7th Framework Programme for research of the European triangulations of A, which are the vertices of the secondary Commission, under FET-Open grant number: 255827. polytope Σ(A). ∗∗The research leading to these results has received fund- ing from the European Union's 7th Framework Programme The Newton polytope of a polynomial is the convex hull (FP7-REGPOT-2009-1) under grant agreement no 245749; of its support, i.e. the exponent vectors of monomials with most of the work in this paper was done at University of nonzero coefficient. It subsumes the notion of degree for Athens, supported by \Computational Geometric Learning". sparse multivariate polynomials by providing more infor- mation, unless the polynomial is completely dense. Given n + 1 polynomials in n variables, with fixed supports Ai and symbolic coefficients, their sparse (or toric) resultant R Permission to make digital or hard copies of all or part of this work for is a polynomial in these coefficients which vanishes exactly personal or classroom use is granted without fee provided that copies are when the polynomials have a common root (Def. 1). The re- not made or distributed for profit or commercial advantage and that copies sultant is the most fundamental tool in elimination theory, bear this notice and the full citation on the first page. To copy otherwise, to and is instrumental in system solving; it is also important republish, to post on servers or to redistribute to lists, requires prior specific in changing representation of parametric hypersurfaces. permission and/or a fee. SCG’12, June 17–20, 2012, Chapel Hill, North Carolina, USA. The Newton polytope of the resultant N(R), or resultant Copyright 2012 ACM 978-1-4503-1299-8/12/06 ...$10.00. polytope, is the object of our study, especially when some of 179 the input coefficients are not symbolic, in which case we seek two univariate polynomials with k0 + 1; k1 + 1 monomials, k0+k1 a projection of the resultant polytope. The lattice points in has vertices and, when both ki ≥ 2, it has k0k1 + 3 k0 N(R) yield a superset of the support of R; this reduces facets. In [29, Sec.6] is proven that N(R) is 1-dimensional implicitization [14, 30] and computation of R to sparse in- iff jAij = 2, for all i, the only planar N(R) is the trian- terpolation (Sect. 2). The number of coefficients of the n+1 gle, whereas the only 3-dimensional ones are the tetrahe- polynomials ranges from O(n) to O(nddn), where d bounds dron, the square-based pyramid, and the polytope of two their total degree. In system solving and implicitization, one univariate trinomials; we compute an instance of the latter computes R when all but O(n) of the coefficients are spe- (Fig. 2(b)). Following [29, Thm.6.2], the 4-dimensional poly- cialized to constants, hence the need for resultant polytope topes include the 4-simplex, some N(R) obtained by pairs of projections. univariate polynomials, and those of 3 trinomials, which we The resultant polytope is a Minkowski summand of Σ(A). can investigate with our code: the maximal such polytope For its construction, we exploit an equivalence relation de- we have computed has f-vector (22; 66; 66; 22) (Fig. 2(c)). fined on the secondary vertices, where the classes are in bi- In [26] they describe all Minkowski summands of Σ(A). jection with the vertices of the resultant polytope. This In [27] is defined an equivalence class over Σ(A) vertices yields an oracle producing a resultant vertex in a given di- having the same mixed cells. The classes map in a many- rection, thus avoiding to compute Σ(A), which has much to-1 fashion to resultant vertices; our algorithm exploits a more vertices than N(R). Although there exist efficient stronger equivalence relationship. Tropical geometry is a software for Σ(A) [28], it is useless in computing resultant polyhedral analogue of algebraic geometry which gives al- polytopes. For instance, in implicitizing parametric surfaces ternative ways of recovering resultant polytopes [21] and with < 100 input terms, we compute the Newton polytope Newton polytopes of implicit equations [30]. Sect. 5 dis- of the equations in < 1sec, which includes all common in- cusses it and shows our approach is faster up to dimensions stances in geometric modeling, whereas Σ(A) is intractable. 5; 6 and competitive in higher dimensions. Our main contribution is twofold. First, we design an We focus on sequences of determinantal predicates. For output-sensitive algorithm for computing the Newton poly- determinants, the record bit complexity is O(n2:697) [23]. tope of R, or of specializations of R. The algorithm com- Methods exist for the sign of general determinants, e.g. [6]. putes both vertex (V) and halfspace (H) representations, We compared linear algebra libraries LinBox [11] and Eigen which is important for the targeted applications. Its incre- [19], which seem most suitable in dimension > 100 and mental nature implies that we also obtain a triangulation medium-high dimensions, respectively, whereas CGAL pro- of the polytope, which may be useful for enumerating its vides the most efficient determinant computation for the di- lattice points in subsequent applications. The complexity mensions to which we focus. is proportional to the number of output vertices and facets; The roadmap of the paper follows: Sect. 2 describes the the overall cost is dominated by computing as many regular combinatorics of resultants, and the following section presents triangulations of A (Thm. 10). We work in the space of the our algorithm. Sect. 4 overcomes the bottleneck of Orienta- projected N(R) and revert to the high-dimensional space of tion predicates. Sect. 5 discusses the implementation, exper- Σ(A) only if needed. Our algorithm readily extends to com- iments, and comparison with other software. We conclude puting Σ(A), projections of Σ(A) and, more generally, any with future work. polytope that can be efficiently described by a vertex ora- cle. A variant of our algorithm computes successively tighter inner and outer approximations: typically, these polytopes 2. RESULTANT POLYTOPES AND THEIR have, respectively, 90% and 105% of the true volume, while PROJECTIONS runtime is reduced up to 25 times. This may lead to an We introduce tools from combinatorial geometry [25, 31] approximation algorithm. to describe resultants [9, 18]. Let vol(·) 2 N denote normal- Second, we describe an efficient implementation based on m × ized Euclidean volume, and (R ) the linear m-dimensional CGAL [7] and the experimental package triangulation. functionals. Our method computes instances of 5-, 6- or 7-dimensional d Let A ⊂ R be a pointset whose convex hull is of dimen- polytopes with 35K, 23K or 500 vertices, respectively, in jAj sion d. For any triangulation T of A, define vector φT 2 R < 2hr. Our code is faster up to dimensions 5 or 6, and com- with coordinate petitive in higher dimensions, to a method computing N(R) X via tropical geometry and based on the Gfan library [21]. φT (a) = vol(σ); a 2 A; (2) Moreover, our code in the critical step of computing convex σ2T :a2σ hulls, uses triangulation which compared to state-of-the- summing over all simplices σ of T having a as a vertex; Σ(A) art software lrs, cdd, and polymake, is the fastest together w is the convex hull of φT for all triangulations T .
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