3412     3241                               4213                           4132                                         41...
PRIMITIVE SORTING NETWORKS           —&— PSEUDOLINE ARRANGEMENTS
PRIMITIVE SORTING NETWORKSnetwork N = n horizontal levels and m vertical commutatorsbricks of N = bounded cells
PSEUDOLINE ARRANGEMENTS ON A NETWORKpseudoline = abscissa-monotone pathcrossing =                       contact =pseudolin...
CONTACT GRAPH OF A PSEUDOLINE ARRANGEMENTcontact graph Λ# of a pseudoline arrangement Λ = • a node for each pseudoline of ...
FLIPSflip = exchange an arbitrary contact with the corresponding crossing           Combinatorial and geometric properties ...
POINT SETS          —&—MINIMAL SORTING NETWORKS
MINIMAL SORTING NETWORKSbubble sort                     insertion sort                         even-odd sort              ...
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKS
POINT SETS & MINIMAL SORTING NETWORKSn points in R2 =⇒ minimal primitive sorting network with n levels                    ...
POINT SETS & MINIMAL SORTING NETWORKS  n points in R2 =⇒ minimal primitive sorting network with n levelsnot all minimal pr...
POINT SETS & MINIMAL SORTING NETWORKSJ. Goodmann & R. Pollack, On the combinatorial classification of nondegenerate configur...
TRIANGULATIONS            —&—ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS
TRIANGULATIONS & ALTERNATING SORTING NETWORKS     triangulation of the n-gon    ←→   pseudoline arrangement               ...
FLIPS
PROPERTIES OF THE FLIP GRAPHThe diameter of the graph of flips on triangulations of the n-gon         is precisely 2n − 10 ...
ASSOCIAHEDRA
PSEUDOTRIANGULATIONS       —&— MULTITRIANGULATIONS
PSEUDOTRIANGULATIONS
PSEUDOTRIANGULATIONS
PSEUDOTRIANGULATIONS
PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P
PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P                        ...
PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P                        ...
PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P                        ...
PSEUDOTRIANGULATIONS                      The flip graph on               pseudotriangulations of a planar                 ...
MULTITRIANGULATIONS
MULTITRIANGULATIONS
MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges                             ...
MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges                             ...
MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges                             ...
BRICK POLYTOPE
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPE  Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn        ω(Λ)j = number of bricks of N bel...
BRICK POLYTOPEXm = network with two levels and m commutatorsgraph of flips G(Xm) = complete graph Km                       ...
BRICK POLYTOPEXm = network with two levels and m commutatorsgraph of flips G(Xm) = complete graph Km                       ...
ASSOCIAHEDRA   —&—PERMUTAHEDRA
ALTERNATING NETWORKS & ASSOCIAHEDRAtriangulation of the n-gon    ←→   pseudoline arrangement                   triangle   ...
ALTERNATING NETWORKS & ASSOCIAHEDRAfor x ∈ {a, b}n−2, define a reduced alternating network Nx and a polygon Px5            ...
ALTERNATING NETWORKS & ASSOCIAHEDRAFor any word x ∈ {a, b}n−2, the brick polytope Ω(Nx ) is an associahedron              ...
DUPLICATED NETWORKS & PERMUTAHEDRAreduced network = network with n levels and n commutators                               ...
DUPLICATED NETWORKS & PERMUTAHEDRAAny pseudoline arrangement supported by Π has one contact and one crossing amongeach pai...
DUPLICATED NETWORKS & PERMUTAHEDRA                              4321                3421            4231            4312  ...
THANK YOU
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AlgoPerm2012 - 09 Vincent Pilaud

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Vincent Pilaud (LIX, CNRS)
Permutahedra, Associahedra and Sorting Networks

Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/

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Transcript of "AlgoPerm2012 - 09 Vincent Pilaud"

  1. 1. 3412 3241 4213 4132 4123 2413 3142 3214 1423 3124 2143 231443 1324 2134 PERMUTAHEDRA, 1234 ASSOCIAHEDRA & SORTING NETWORKS Vincent PILAUD
  2. 2. PRIMITIVE SORTING NETWORKS —&— PSEUDOLINE ARRANGEMENTS
  3. 3. PRIMITIVE SORTING NETWORKSnetwork N = n horizontal levels and m vertical commutatorsbricks of N = bounded cells
  4. 4. PSEUDOLINE ARRANGEMENTS ON A NETWORKpseudoline = abscissa-monotone pathcrossing = contact =pseudoline arrangement (with contacts) = n pseudolines supported by N which havepairwise exactly one crossing, possibly some contacts, and no other intersection
  5. 5. CONTACT GRAPH OF A PSEUDOLINE ARRANGEMENTcontact graph Λ# of a pseudoline arrangement Λ = • a node for each pseudoline of Λ, and • an arc for each contact of Λ oriented from top to bottom
  6. 6. FLIPSflip = exchange an arbitrary contact with the corresponding crossing Combinatorial and geometric properties of the graph of flips G(N )? VP & M. Pocchiola, Multitriangulations, pseudotriangulations and sorting networks, 2012+ VP & F. Santos, The brick polytope of a sorting network, 2012 A. Knutson & E. Miller, Subword complexes in Coxeter groups, 2004 C. Ceballos, J.-P. Labb´ & C. Stump, Subword complexes, cluster complexes, and generalized multi-associahedra, 2012+ e VP & C. Stump, Brick polytopes of spherical subword complexes [. . . ], 2012+
  7. 7. POINT SETS —&—MINIMAL SORTING NETWORKS
  8. 8. MINIMAL SORTING NETWORKSbubble sort insertion sort even-odd sort D. Knuth, The art of Computer Programming (Vol. 3, Sorting and Searching), 1997
  9. 9. POINT SETS & MINIMAL SORTING NETWORKS
  10. 10. POINT SETS & MINIMAL SORTING NETWORKS
  11. 11. POINT SETS & MINIMAL SORTING NETWORKS
  12. 12. POINT SETS & MINIMAL SORTING NETWORKS
  13. 13. POINT SETS & MINIMAL SORTING NETWORKS
  14. 14. POINT SETS & MINIMAL SORTING NETWORKS
  15. 15. POINT SETS & MINIMAL SORTING NETWORKS
  16. 16. POINT SETS & MINIMAL SORTING NETWORKS
  17. 17. POINT SETS & MINIMAL SORTING NETWORKS
  18. 18. POINT SETS & MINIMAL SORTING NETWORKSn points in R2 =⇒ minimal primitive sorting network with n levels point ←→ pseudoline edge ←→ crossing boundary edge ←→ external crossing
  19. 19. POINT SETS & MINIMAL SORTING NETWORKS n points in R2 =⇒ minimal primitive sorting network with n levelsnot all minimal primitive sorting networks correspond to points sets of R2 =⇒ realizability problems
  20. 20. POINT SETS & MINIMAL SORTING NETWORKSJ. Goodmann & R. Pollack, On the combinatorial classification of nondegenerate configurations in the plane, 1980 D. Knuth, Axioms and Hulls, 1992 A. Bj¨rner, M. Las Vergnas, B. Sturmfels, N. White, & G. Ziegler, Oriented Matroids, o 1999 J. Bokowski, Computational oriented matroids, 2006
  21. 21. TRIANGULATIONS —&—ALTERNATING SORTING NETWORKS
  22. 22. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  23. 23. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  24. 24. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  25. 25. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  26. 26. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  27. 27. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  28. 28. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  29. 29. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  30. 30. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  31. 31. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  32. 32. TRIANGULATIONS & ALTERNATING SORTING NETWORKS triangulation of the n-gon ←→ pseudoline arrangement triangle ←→ pseudoline edge ←→ contact point common bisector ←→ crossing point dual binary tree ←→ contact graph
  33. 33. FLIPS
  34. 34. PROPERTIES OF THE FLIP GRAPHThe diameter of the graph of flips on triangulations of the n-gon is precisely 2n − 10 when n is large enough. D. Sleator, R. Tarjan, & W. Thurston, Rotation distance, triangulations, and hyperbolic geometry, 1988The graph of flips on triangulations of the n-gon is Hamiltonian. L. Lucas, The rotation graph of binary trees is Hamiltonian, 1988 F. Hurado & M. Noy, Graph of triangulations of a convex polygon and tree of triangulations, 1999 The graph of flips on triangulations of the n-gon is polytopal. C. Lee, The associahedron and triangulations of the n-gon, 1989 L. Billera, P. Filliman, & B. Strumfels, Construction and complexity of secondary polytopes, 1990 J.-L. Loday, Realization of the Stasheff polytope, 2004 C. Holhweg & C. Lange, Realizations of the associahedron and cyclohedron, 2007 A. Postnikov, Permutahedra, associahedra, and beyond, 2009 VP & F. Santos, The brick polytope of a sorting network, 2012 C. Ceballos, F. Santos, & G. Ziegler, Many non-equivalent realizations of the associahedron, 2012+
  35. 35. ASSOCIAHEDRA
  36. 36. PSEUDOTRIANGULATIONS —&— MULTITRIANGULATIONS
  37. 37. PSEUDOTRIANGULATIONS
  38. 38. PSEUDOTRIANGULATIONS
  39. 39. PSEUDOTRIANGULATIONS
  40. 40. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P
  41. 41. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P = complex of pseudotriangles
  42. 42. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P = complex of pseudotrianglesobject from computational geometryapplications to visibility, rigidity, motion planning, . . .
  43. 43. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P = complex of pseudotrianglesobject from computational geometryapplications to visibility, rigidity, motion planning, . . .properties of the flip graph: Ω(n) ≤ diameter ≤ O(n ln n) graph of the pseudotriangulation polytope
  44. 44. PSEUDOTRIANGULATIONS The flip graph on pseudotriangulations of a planar point set P is polytopal G. Rote, F. Santos, I. Streinu, Expansive motions and the polytope of pointed pseudotriangulations, 2008
  45. 45. MULTITRIANGULATIONS
  46. 46. MULTITRIANGULATIONS
  47. 47. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
  48. 48. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges = complex of k -stars
  49. 49. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges = complex of k -starsobject from combinatoricscounted by the Hankel determinant det([Cn−i−j ]1≤i,j≤n) of Catalan numbers, . . .
  50. 50. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges = complex of k -starsobject from combinatoricscounted by the Hankel determinant det([Cn−i−j ]1≤i,j≤n) of Catalan numbers, . . .properties of the flip graph: (k + 1/2)n ≤ diameter ≤ 2kn graph of a combinatorial sphere
  51. 51. BRICK POLYTOPE
  52. 52. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of ΛBrick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  53. 53. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of Λ 2Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  54. 54. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of Λ 6 2Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  55. 55. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of Λ 8 6 2Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  56. 56. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of Λ 1 8 6 2Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  57. 57. BRICK POLYTOPE Λ pseudoline arrangement supported by N −→ brick vector ω(Λ) ∈ Rn ω(Λ)j = number of bricks of N below the j th pseudoline of Λ 6 1 8 6 2Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }
  58. 58. BRICK POLYTOPEXm = network with two levels and m commutatorsgraph of flips G(Xm) = complete graph Km m−i m−1 0brick polytope Ω(Xm) = conv i ∈ [m] = , i−1 0 m−1
  59. 59. BRICK POLYTOPEXm = network with two levels and m commutatorsgraph of flips G(Xm) = complete graph Km m−i m−1 0brick polytope Ω(Xm) = conv i ∈ [m] = , i−1 0 m−1 The brick vector ω(Λ) is a vertex of Ω(N ) ⇐⇒ the contact graph Λ# is acyclic The graph of the brick polytope Ω(N ) is a subgraph of the flip graph G(N ) The graph of the brick polytope Ω(N ) coincides with the graph of flips G(N ) ⇐⇒ the contact graphs of the pseudoline arrangements supported by N are forests
  60. 60. ASSOCIAHEDRA —&—PERMUTAHEDRA
  61. 61. ALTERNATING NETWORKS & ASSOCIAHEDRAtriangulation of the n-gon ←→ pseudoline arrangement triangle ←→ pseudoline edge ←→ contact point common bisector ←→ crossing point dual binary tree ←→ contact graph The brick polytope is an associahedron.
  62. 62. ALTERNATING NETWORKS & ASSOCIAHEDRAfor x ∈ {a, b}n−2, define a reduced alternating network Nx and a polygon Px5 5 54 a 4 a 4 a3 a 3 a 3 b2 a 2 b 2 a1 1 1 2 3 4 2 3 2 41 a a a 5 1 a a b 5 1 a b a 5 4 3 1 Pseudoline arrangements on Nx ←→ triangulations of the polygon Px.
  63. 63. ALTERNATING NETWORKS & ASSOCIAHEDRAFor any word x ∈ {a, b}n−2, the brick polytope Ω(Nx ) is an associahedron 1 C. Hohlweg & C. Lange, Realizations of the associahedron and cyclohedron, 2007 VP & F. Santos, The brick polytope of a sorting network, 2012
  64. 64. DUPLICATED NETWORKS & PERMUTAHEDRAreduced network = network with n levels and n commutators 2 it supports only one pseudoline arrangementduplicated network Π = network with n levels and 2 n commutators obtained by 2 duplicating each commutator of a reduced network Any pseudoline arrangement supported by Π has one contact and one crossing among each pair of duplicated commutators.
  65. 65. DUPLICATED NETWORKS & PERMUTAHEDRAAny pseudoline arrangement supported by Π has one contact and one crossing amongeach pair of duplicated commutators =⇒ The contact graph Λ# is a tournament. Vertices of Ω(Π) ⇐⇒ acyclic tournaments ⇐⇒ permutations of [n] Brick polytope Ω(Π) = permutahedron
  66. 66. DUPLICATED NETWORKS & PERMUTAHEDRA 4321 3421 4231 4312 3412 2431 3241 4132 4213 2341 4123 2413 3142 1432 3214 1342 1423 3124 2143 2314 1243 1324 2134 1234
  67. 67. THANK YOU
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