For physical models of polyhedra, see Polyhedron model.
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to be explicitly enumerated -- thereby requiring a compact representation. Nested loop programs are the typical, but not the only example, and the most common use of the model is for loop nest optimization in program optimization. The polyhedral method treats each loop iteration within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations such as tiling on the polytopes, and then converts the transformed polytopes into equivalent, but optimized (depending on targeted optimization goal), loop nests through polyhedra scanning.
The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large...
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional...
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any...
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In...
regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. Only counting polytopes of...
octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan (ℓ1) metric. If the...
Use of the polyhedral model (also called the polytopemodel) within a compiler requires software to represent the objects of this framework (sets of integer-valued...
examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Convex polyhedra are...
research as of the time of this writing (2010). Loop nest optimization Polytopemodel Scalable parallelism Scalable locality In the book Reasoning About Program...
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset...
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries...
regular polytopes. Coxeter lists a few of these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals:...
compiler Loop nest optimization Parallelization contract Polytopemodel also known as Polyhedral model Scalable parallelism BMDFM Vectorization SequenceL Yehezkael...
single plane. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons...
affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or...
tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme...
of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the...
S2CID 250764497. Roth, J. (2007-10-24). "Description of a highly symmetric polytope observed in Thomson's problem of charges on a hypersphere". Physical Review...
Xenakis's UPIC system; and the massive multimedia performances Xenakis called polytopes, that were a summa of his interests and skills. Among the numerous theoretical...
broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Take some corner or vertex of a polyhedron. Mark a point...
polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure...
on the comparison mensuration. It also has many relations with other polytopes. The appearance of regular icosahedron can be found in nature, such as...
of all the regions. Every region turns out to geometrically be a convex polytope for linear MPC, commonly parameterized by coefficients for its faces, requiring...
a suggestion from Albert Einstein. 120-cell – a regular polychoron (4D polytope) whose surface consists of 120 dodecahedral cells Braarudosphaera bigelowii...
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