We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B 11 ,. .. , B 15 , whereas exact methods have only computed that of B 10 .
We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex-and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6-or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.
Aim To assess contemporary and historical determinants of taxonomic and ecological trait turnover in birds worldwide. We tested whether taxonomic and trait turnover (1) are structured by regional bioclimatic conditions, (2) increase in relationship with topographic heterogeneity and environmental turnover and change according to current and historical environmental conditions, and (3) decrease with human impact. Major Taxa Birds. Location Global. Methods We used computationally efficient algorithms to map the taxonomic and trait turnover of 8,040 terrestrial bird assemblages worldwide, based on a grid with 110 km × 110 km resolution overlaid on the extent‐of‐occurrence maps of 7,964 bird species, and nine ecological traits reflecting six key aspects of bird ecology (diet, habitat use, thermal preference, migration, dispersal and body size). We used quantile regression and model selection to quantify the influence of biomes, environment (temperature, precipitation, altitudinal range, net primary productivity, Quaternary temperature and precipitation change) and human impact (human influence index) on bird turnover. Results Bird taxonomic and trait turnover were highest in the north African deserts and boreal biomes. In the tropics, taxonomic turnover tended to be higher, but trait turnover was lower than in other biomes. Taxonomic and trait turnover exhibited markedly different or even opposing relationships with climatic and topographic gradients, but at their upper quantiles both types of turnover decreased with increasing human influence. Main conclusions The influence of regional, environmental and anthropogenic factors differ between bird taxonomic and trait turnover, consistent with an imprint of niche conservatism, environmental filtering and topographic barriers on bird regional assemblages. Human influence on these patterns is pervasive and demonstrates global biotic homogenization at a macroecological scale.
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 α−n−1, where the input consists of α 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.
A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d 7. Our approach is inductive: for each fixed (d − 1)-dimensional 2-level polytope P 0 , we enumerate all d-dimensional 2-level polytopes P that have P 0 as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet P 0 , we obtain all 2-level polytopes in dimension d.(1,3,4,5) Université libre de Bruxelles, Brussels, Belgium Lemma 4. Let P be a d-polytope having facet-defining inequalities g 1 (x) 0, . . . , g m (x) 0, and vertices v 1 , . . . , v n . If σ denotes a map from the affine hull aff(P ) of P to R m defined by σ(x) i := g i (x) for all x ∈ aff(P ), then the polytopes P and σ(P ) are affinely equivalent.Proof. The map σ : aff(P ) → R m is affine, and injective because it maps the vertices of any simplicial core for P to affinely independent points. The result follows.By definition, a polytope P is 2-level if and only if S(P ) can be scaled to be a 0/1 matrix. Given a 2-level polytope, we henceforth always assume that its facet-defining inequalities are scaled so that the slacks are 0/1. Thus, the slack embedding of a 2-level polytope depends only on the support 1 of its slack matrix, which only depends on its combinatorial structure. As a consequence, we have the following result:Lemma 5. Two 2-level polytopes are affinely equivalent if and only if they have the same combinatorial type.
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