2. An Algorithm for Polytope Decomposition and Exact Computation of Multiple Integrals

3. Overview Definitions Background Some algorithmic problems in polytope theory Repetitive decomposition of a polyhedron Calculating multiple integrals (and volumes) Uniformly repetitive decomposition of polyhedra finding distribution functions

4. Definition Polyhedron Examples of bounded and unbounded polyhedra

5. Polytope A Polytope is a bounded polyhedron.

6. H-V representation of polytope V-RepresentationH-Representation

7. Simplex A Simplex has vertices.

8. Triangulation (a) A triangulation using 6-simplices (b) A triangulation using 5-simplices

9. Boundary triangulation

10. Signed Decomposition Methods

11. Volume of simplex be the vertices ofLeta -simplex The volume of the simplex is:

12. Some algorithmic problems in polytope theory Number of vertices Input: Polytope in -representation Output: Number of vertices of Status (general): -complete Status (fixed dim.): Polynomial time

13. Some algorithmic problems in polytope theory (cont.) Minimum triangulation Input: Polytope in -representation, positive integer k Output: “ Yes ” if has a triangulation of size k or less, “ No ” otherwise Status (general): -complete Status (fixed dim.): -complete

14. Minimum Triangulation A polygon has simplices minimum

15. Minimum Triangulation (a)A triangulation using 6-simplices (b) A triangulation using 5-simplices

16. Some algorithmic problems in polytope theory (cont.) Volume Input: Polytope in -representation, Output: Volume of P Status (general): -complete Status (fixed dim.): Polynomial time

17. Repetitive decomposition of a polyhedron

18. Definition of a repetitive polyhedron A polytope is repetitive if it may be represented in the form for appropriate and linear functions

19. Example of a repetitive polyhedron

20. Theorem 1 Theorem 1: Any polyhedron P is effectively decomposable into a union of finitely many repetitive polyhedra, the intersection of any two of which is contained in a -dimensional polytope.

21. Proof of Theorem 1.

22. Proof of Theorem 1.(cont.)

25. Decomposition into repetitive polytopes Y X

26. Decomposition into repetitive polytopes (cont.) Y X

27. Y X

28. Decomposition into repetitive polyhedra

29. Multiple integral for repetitive polyhedron

30. Multiple integral

31. Volume of a repetitive polytope

32. Uniformly repetitive decomposition of polyhedra

33. Background Let be a -dimensional random variable, uniformly distributed in the polytope. That is, the probability of to assume a value in some set is.

34. Background(cont.) Consider a 1-dimensional random variable of the form for some constants. Then the value of the distribution function at any point t is.

35. Classical example Let, where is uniformly distributed in the d - dimensional cube. That is, is the sum of d independent variables distributed uniformly in.For example,

36. Classical example

37. Example Let We would like to express as a function of.

38. Example(cont.)

39. Definition of uniformly repetitive polyhedra Let be a family of polyhedra, where is some interval (finite or infinite). The family is uniformly repetitive if there exist linear functions, such that (where some of the functions or may be replaced by or ).

40. Example of decomposition into uniformly repetitive families(cont.)

41. Example of decomposition into uniformly repetitive families

42. Result of decomposition

43. Theorem 2:( ) Let be a polyhedron and a linear function. Then we can effectively find a decomposition of, say, into a union of finitely many (finite and infinite ) intervals, and uniformly repetitive families, such that

44. Proof of Theorem 2.

45. Proof of Theorem 2.(cont.)

47. Example of decomposition into uniformly repetitive families(cont.)

48. Example of decomposition into uniformly repetitive families

49. Result of decomposition

50. EXAMPLE (CONT.) Distribution function

51. Theorem 3: Let be a -dimensional random variable, uniformly distributed in a polytope of positive volume in. Given any constants, the distribution function of the 1-dimensional random variable is a continuous piecewise polynomial function of the degree at most, and can be effectively computed.

52. Distribution function

53. Polytope decomposition

54. Questions & Answers