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A new definition for the dynamics

Explore a novel approach to graph theory dynamics with a focus on minima, separation, and ordered minima. Understand pass values, k-separation, and the impact on dynamics. Discover how dynamics and separations are interconnected.

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A new definition for the dynamics

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  1. A new definition for the dynamics G. Bertrand Laboratoire A2SI, ESIEE Institut Gaspard Monge – UMR UMLV/ESIEE/CNRS ISMM 2005

  2. A discrete approach • Let G = (V,E) be an (undirected) graph. • We denote by Func (V) the family composed of all maps from V to Z. ISMM 2005

  3. Pass value • Let F be in Func (V). If п is a path, we set F(п) = Max{F(x); x  п}. • Let x, y in V. We set F(x,y) = Min {F(п); п  п(x,y)}, F(x,y) is the pass value between x and y. • Let X and Y be two subsets of V. We set F(X,Y) = Min{F(x,y); x  X and y  Y}. ISMM 2005

  4. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  5. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  6. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  7. Pass value 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005 F(X,Y) = 31

  8. Dynamics (M. Grimaud,1992) • Let X be a minimum for F. Let G(X) be the number such that: i) if X = Xmin, then G(X) = infinity; ii) otherwise, G(X) = Min {F(X,Y); for all minima Y such that F(Y) < F(X)}. • The dynamics of a minimum X is the number Dyn(X) = G(X) – F(X) ISMM 2005

  9. Dynamics ISMM 2005

  10. Dynamics ∞ ISMM 2005

  11. Dynamics ∞ ISMM 2005

  12. Dynamics ∞ ISMM 2005

  13. Dynamics ∞ ISMM 2005

  14. Dynamics ∞ ISMM 2005

  15. Dynamics ∞ ISMM 2005

  16. k-Separation • Let F be in Func (V) and let x and y be in V. We say that x and y are separated (for F) if F(x,y) > Max{F(x),F(y)}.We say that x and y are k-separated (for F) if x and y are separated and F(x,y) = k. ISMM 2005

  17. y x k-separation x and y are not separated 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  18. y x k-separation x and y are 20-separated 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  19. Separation • Let F and G be in Func (V) such that G  F.We say that G is a separationof F if, for all x,y in V, if x and y are k-separated for F, then x and y are k-separated for G. ISMM 2005

  20. Separation F ISMM 2005 G

  21. Separation F ISMM 2005 G

  22. Separation F K ISMM 2005 G

  23. Separation F ISMM 2005 G

  24. Separation F K ISMM 2005 G

  25. Separation F K ISMM 2005 G G is a separation of F

  26. Dynamics and separation Let G ≤ F (G being a minima extension of F) If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F ISMM 2005

  27. Dynamics and separation Let G ≤ F (G being a minima extension of F). If G is a separation of F, then the dynamics of a minimum of G is the same than the dynamics of the corresponding minimum of F. The converse is not true ISMM 2005

  28. Dynamics: counter-example F ∞ ISMM 2005

  29. Dynamics: counter-example G ∞ ISMM 2005

  30. Ordered minima • Let F be in F (V). A minima ordering (for F) is a strict total order relation < on the minima of F. • Let X be a minimum for F. The pass valueof X for (F,<) is the number F(X,<) such that: i) if X = Xmin, then F(X,<) = infinity; ii) otherwise, F(X,<) = Min {F(X,Y); for all minima Y such that Y < X}. ISMM 2005

  31. Ordered minima 40 40 40 40 40 40 40 40 40 40 40 40 40 40 1 1 2 3 10 5 25 5 4 4 4 40 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  32. Ordered minima F(.,<)=8 40 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 0 2 8 6 5 5 20 3 2 3 40 F(.,<)=20 40 3 3 2 3 10 6 6 6 22 2 3 F(.,<)=30 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 F(.,<)=infty F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  33. Ordered dynamics • The notion of ordered pass values leads to a new definition of the dynamics of a minimum: Dyn(X; F, <) = F(X, <) – F(X) ISMM 2005

  34. Ordered minima Dyn(.,<)=8-5 40 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 0 2 8 6 5 5 20 3 2 3 40 40 3 3 2 3 10 6 6 6 22 2 3 40 Dyn(.,<)=20-0 40 6 6 40 6 11 11 11 25 4 4 4 40 Dyn(.,<)=30-2 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 Dyn(.,<)=31-3 40 40 40 40 40 40 40 40 40 40 40 40 40 Dyn(.,<)=infty ISMM 2005

  35. Theorem(ordered dynamics and separation) Let G ≤ F (G being a minima extension of F). Let < be a minima ordering for F. The map G is a separation of F if and only if, for each minimum X for F, we have Dyn(X; F, <) = Dyn(X; G, <) . ISMM 2005

  36. Dynamics: counter-example ∞ ISMM 2005

  37. Ordered minima F ISMM 2005

  38. Ordered minima F 2 3 ISMM 2005 1

  39. Ordered minima ∞ F 2 3 ISMM 2005 1

  40. Ordered minima ∞ F 2 3 ISMM 2005 1

  41. Ordered minima ∞ G 2 3 ISMM 2005 1

  42. Ordered minima F 1 2 ISMM 2005 3

  43. Ordered minima ∞ F 1 2 ISMM 2005 3

  44. Ordered minima ∞ F 1 2 ISMM 2005 3

  45. Ordered minima ∞ G 1 2 ISMM 2005 3

  46. Remark If all the minima of a function F are distinct and if the ordering of the minima of F is made according to the altitudes of the minima of F, then the ordered dynamics of a minimum is equal to the unordered dynamics of this minimum. ISMM 2005

  47. A tree associated to F and < F(.,<)=8 40 40 40 40 40 40 40 40 40 40 40 40 40 5 40 3 1 1 2 3 10 5 25 5 4 4 4 40 2 40 1 0 2 8 6 5 5 20 3 2 3 40 F(.,<)=20 40 3 3 2 3 10 6 6 6 22 2 3 F(.,<)=30 40 40 6 6 40 6 11 11 11 25 4 4 4 40 40 40 35 10 30 15 15 15 35 31 36 10 40 40 10 8 5 10 32 33 34 10 10 15 38 40 1 4 40 8 5 1 1 15 40 10 6 3 15 20 40 40 10 8 5 10 15 35 15 6 6 15 35 40 F(.,<)=0 F(.,<)=31 40 40 40 40 40 40 40 40 40 40 40 40 40 ISMM 2005

  48. Theorem(minimum spanning tree) Let F be in F (V) and let < be a minima ordering for F. Let T be a tree associated to F and <. Let G’ be the complete graph the vertices of which are the minima of F, an edge being labeled by the corresponding pass value. The tree T is a minimum spanning tree of G’. ISMM 2005

  49. Conclusion ISMM 2005 Dyn > 22

  50. Conclusion ISMM 2005 => Ordering the minima with arbitary criteria

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