( Shelling (Bruggesser-Mani 1971) and Ranking Let be a polytope. has such a representation iff it contains the origin in its interior. For a generic, sort the inequalities so that. a ranking of vertices s in the dual by a linear function). Geometrically, the line meets each hyperplane. Let denotes the parameter value at the intersection. Thus, and Consequently: This ordering induces a shelling of "#! $ # ' & is a topological. %& : -ball for each 24
Shelling and Ranking (cont.) c z 2 F 2 F 1 F 3 P z 1 0 z 5 F 5 z 4 F 4 z 3 25
Shelling in (Launching a Space Shuttle) z 5 L z 3 z 4 z 1 z 12 z 2 (a) z 4 F 4 (b) 26
Shelling in (cont.) F 3 F 1 F 2 F 5 F 6 F 4 F 9 F 7 F 8 F 11 F 12 F 10 27
The Double Description Method: Complexity? : an H-polytope represented by halfspaces,, in. $ " : ' th polytope ( ). : the vertex set computed at ' th step. (P k-1, V k-1 ) (P k, V k ) h k newly generated for each adj pair (, ) 28
The Double Description Method: Complexity? (cont.) It is an incremental method, dual to the Beneath-Beyond Method. Practical for low dimensions and highly degenerate inputs. For highly degenerate inputs, the sizes of intermediate polytopes are very sensitive to the ordering of halfspaces. For example, the maxcutoff ordering ( the deepest cut ) may provoke extremely high intermediate sizes. It is hard to estimate its complexity in terms of and the sizes of input and output. The main reason is that the intermediate polytopes can become very complex relative to the original polytope. D. Bremner (1999) proved that there is a class of polytopes for which the double description method (and the beneath-beyond) method is exponential. 29
How Intermediate Sizes Fluctuate with Different Orderings Size INTERMEDIATE SIZES FOR CCP6 1500 1250 random maxcutof mincutoff 1000 750 500 250 lexmin 20 22 24 26 28 30 32 Iteration The input is a -dimensional polytope with facets. The output is a list of vertices. The lexmin is a sort of shelling ordering. 30
How Intermediate Sizes Fluctuate with Different Orderings Size 30000 Maxcutoff 25000 20000 Random 15000 10000 5000 Mincutoff 200 400 600 800 1000 Iteration The input is a -dimensional cross polytope with facets. The output is a list of vertices. The highest peak is attained by maxcutoff ordering, following by random and mincutoff. Lexmin is the best among all and the peak intermediate size is less than. (Too small too see it above.) 31
Pivoting Algorithms for Vertex Enumeration Basic Idea: Search the connected graph of an H-polytope operations to list all vertices. by pivoting A polytope and its graph (1-skeleton) Advantage: Under the usual nondegeneracy (i.e. no points in lie on more than facets), it is polynomial in the input size and the output size. % Space Complexity: Depends on the search technique. The standard depth-first search requires to store all vertices found. 32
Memory Free: Reverse Search for Vertex Enumeration Key idea: Reverse the simplex method from the optimal vertex in all possible ways: min x1 + x2 + x3 $"!! #" "*!$ Complexity: % -time and % 33 -space (under nondegeneracy).
Reverse Search: General Description Two functions and A finite local search is a function: define the search: for a graph satisfying with a special node (L1) for each, and (L2) for each Example:, ' such that. Let be a simple polytope, and be any generic linear objective function. Let be the set of all vertices of, the unique optimal, and be the vertex adjacent to selected by the (deterministic) simplex method. 34
Reverse Search: General Description A adjacency oracle foragraph upper bound for the maximum degree of is a function (where ) satisfying: a (i) for each vertex and each number with, a vertex adjacent to or extraneous (ii) if ' ' ' ' for some ', ' the oracle returns (zero), and ', then ' ', (iii) for each vertex, the set of vertices adjacent to ' '. ' is exactly Example: Let be a simple polytope. Let be the set of all vertices of, be the number of nonbasic variables and be the vertex adjacent to obtained by pivoting on the th nonbasic variable at. ' ' 35
(* restore Reverse Search: General Description (r1) (r2) (f1) procedure ReverseSearch( ); ; (* ; : neighbor counter *) repeat while do ; if if,, then, ; ; then (* reverse traverse *) endif endif endwhile; if then (* forward traverse *) (f2) endif until ; ; repeat and ; until *) 36
Pivoting Algorithm vs Incremental Algorithm Pivoting algorithms, in particular the reverse search algorithm (lrs, lrslib), work well for high dimensional cases. Incremental algorithms work well for low (up to ) dimensional cases and highly degenerate cases. For example, the codes cdd/cddlib and porta are implemented for highly degenerate cases and the code qhull for low (up to ) dimensional cases. The reverse search algorithm seems to be the only method that scales very efficiently in massively parallel environment. Various comparisons of representation conversion algorithms and implementations can be found in the excellent article: D. Avis, D. Bremner, and R. Seidel. How good are convex hull algorithms. Computational Geometry: Theory and Applications, 7:265 302, 1997. 37
Voronoi Diagram in : a set of distinct points in Voronoi diagram is the partition of into polyhedral regions: % % for & where is the Euclidean distance function. Each region the Voronoi cell of. % is called 38
Voronoi Diagram as Polyhedral Projection ( For in, consider the hyperplane ) in at tangent to the paraboloid : " " & Replacing equation with inequality polyhedron for each, we obtain the " " & The key observation is that for two distinct pointsand, the intersection of two hyperplanes and is in fact the equalseparator hyperplane of the two points. 39
Voronoi Diagram as Polyhedral Projection The Voronoi diagram is simply the orthogonal projection of onto the original space. The projected vertices of are called the Voronoi vertices. 40
Delaunay Triangulation in For each point, the nearest neighbor set of points which are closest to in Euclidean distance. is the set of The convex hull vertex is called the Delaunay cell of is a partition of the convex hull. The Delaunay triangulation of into the Delaunay cells of Voronoi vertices. The one-to-one correspondence between the Voronoi vertices and the Delaunay cells is duality that. of the nearest neighbor set of a Voronoi 41
Arrangement of Hyperplanes A finite family an arrangement of hyperplanes. of hyperplanes in is called h 2 h 5 h 4 h 1 h 3 42
! Arrangement of Hyperplanes: Representation of Faces h 1 + h 5 h 2 h 4 _ + + + _ h 3 + (0-0-+) (+++++) (++++0) 43
Central Arrangement of Hyperplanes An arrangement of hyperplanes in which all its hyperplanes contain the origin is called a central arrangement of hyperplanes. h 2 h 4 h 1 + - -+ + - - + h 3 0 44
Central Arrangement and Sphere Arrangement S 1 (-+++) S 2 (00++) S 3 - + - + (-0++) + - S 4 + - 45
Polyhedral Realization of Arrangements Let be an arrangement of hyperplanes represented by a matrix Consider the following polytope:., i.e, & Theorem 0.14. The face lattice of the polytope. is isomorphic to the face lattice of 46
& & Dual of : A Zonotope The polar of the polytope is the polytope (called zonotope) where each generator is the line segment &. (P A ) * 47
& & Dual of : A Zonotope The polar of the polytope is the polytope (called zonotope) where each generator is the line segment &. (P A ) * 48
Examples of Zonotopes in Random zonotopes with and generators: z z x y x y 49