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AlgoPerm2012 - 09 Vincent Pilaud

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Vincent Pilaud (LIX, CNRS) …

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

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

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  • 1. 3412 3241 4213 4132 4123 2413 3142 3214 1423 3124 2143 231443 1324 2134 PERMUTAHEDRA, 1234 ASSOCIAHEDRA & SORTING NETWORKS Vincent PILAUD
  • 2. PRIMITIVE SORTING NETWORKS —&— PSEUDOLINE ARRANGEMENTS
  • 3. PRIMITIVE SORTING NETWORKSnetwork N = n horizontal levels and m vertical commutatorsbricks of N = bounded cells
  • 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. 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. 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. POINT SETS —&—MINIMAL SORTING NETWORKS
  • 8. MINIMAL SORTING NETWORKSbubble sort insertion sort even-odd sort D. Knuth, The art of Computer Programming (Vol. 3, Sorting and Searching), 1997
  • 9. POINT SETS & MINIMAL SORTING NETWORKS
  • 10. POINT SETS & MINIMAL SORTING NETWORKS
  • 11. POINT SETS & MINIMAL SORTING NETWORKS
  • 12. POINT SETS & MINIMAL SORTING NETWORKS
  • 13. POINT SETS & MINIMAL SORTING NETWORKS
  • 14. POINT SETS & MINIMAL SORTING NETWORKS
  • 15. POINT SETS & MINIMAL SORTING NETWORKS
  • 16. POINT SETS & MINIMAL SORTING NETWORKS
  • 17. POINT SETS & MINIMAL SORTING NETWORKS
  • 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. 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. 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. TRIANGULATIONS —&—ALTERNATING SORTING NETWORKS
  • 22. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 23. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 24. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 25. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 26. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 27. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 28. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 29. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 30. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 31. TRIANGULATIONS & ALTERNATING SORTING NETWORKS
  • 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. FLIPS
  • 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. ASSOCIAHEDRA
  • 36. PSEUDOTRIANGULATIONS —&— MULTITRIANGULATIONS
  • 37. PSEUDOTRIANGULATIONS
  • 38. PSEUDOTRIANGULATIONS
  • 39. PSEUDOTRIANGULATIONS
  • 40. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P
  • 41. PSEUDOTRIANGULATIONSpseudotriangulation of P = maximal crossing-free and pointed set of edges on P = complex of pseudotriangles
  • 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. 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. 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. MULTITRIANGULATIONS
  • 46. MULTITRIANGULATIONS
  • 47. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
  • 48. MULTITRIANGULATIONSk -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges = complex of k -stars
  • 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. 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. BRICK POLYTOPE
  • 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. 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. 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. 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. 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. 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. 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. 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. ASSOCIAHEDRA —&—PERMUTAHEDRA
  • 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. 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. 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. 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. 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. 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. THANK YOU

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