The document discusses various topics related to sorting networks and their connections to mathematical structures: 1) It describes primitive sorting networks and their representation as pseudoline arrangements, along with properties of the contact graph and the graph of possible flips between arrangements. 2) It discusses relationships between point sets, minimal sorting networks, and oriented matroids. 3) It covers triangulations, their representation as alternating sorting networks and pseudoline arrangements, and how the brick polytope of these networks gives associahedra. 4) Other topics mentioned include pseudotriangulations, multitriangulations, and how duplicated networks relate to permutahedra.

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PERMUTAHEDRA,
1234 ASSOCIAHEDRA
& SORTING NETWORKS
Vincent PILAUD

2. PRIMITIVE SORTING NETWORKS
—&—
PSEUDOLINE ARRANGEMENTS

3. PRIMITIVE SORTING NETWORKS
network N = n horizontal levels and m vertical commutators
bricks of N = bounded cells

4. PSEUDOLINE ARRANGEMENTS ON A NETWORK
pseudoline = abscissa-monotone path
crossing = contact =
pseudoline arrangement (with contacts) = n pseudolines supported by N which have
pairwise exactly one crossing, possibly some contacts, and no other intersection

5. CONTACT GRAPH OF A PSEUDOLINE ARRANGEMENT
contact 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. FLIPS
flip = 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 NETWORKS
bubble 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 NETWORKS
n 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 levels
not all minimal primitive sorting networks correspond to points sets of R2
=⇒ realizability problems

20. POINT SETS & MINIMAL SORTING NETWORKS
J. 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 GRAPH
The 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, 1988
The 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. PSEUDOTRIANGULATIONS
pseudotriangulation of P = maximal crossing-free and pointed set of edges on P

41. PSEUDOTRIANGULATIONS
pseudotriangulation of P = maximal crossing-free and pointed set of edges on P
= complex of pseudotriangles

42. PSEUDOTRIANGULATIONS
pseudotriangulation of P = maximal crossing-free and pointed set of edges on P
= complex of pseudotriangles
object from computational geometry
applications to visibility, rigidity, motion planning, . . .

43. PSEUDOTRIANGULATIONS
pseudotriangulation of P = maximal crossing-free and pointed set of edges on P
= complex of pseudotriangles
object from computational geometry
applications 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. MULTITRIANGULATIONS
k -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges

48. MULTITRIANGULATIONS
k -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
= complex of k -stars

49. MULTITRIANGULATIONS
k -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
= complex of k -stars
object from combinatorics
counted by the Hankel determinant det([Cn−i−j ]1≤i,j≤n) of Catalan numbers, . . .

50. MULTITRIANGULATIONS
k -triangulation of the n-gon = maximal (k + 1)-crossing-free set of edges
= complex of k -stars
object from combinatorics
counted 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 Λ
Brick 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
2
Brick 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
2
Brick 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
2
Brick 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
2
Brick 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
2
Brick polytope Ω(N ) = conv {ω(Λ) | Λ pseudoline arrangement supported by N }

58. BRICK POLYTOPE
Xm = network with two levels and m commutators
graph of flips G(Xm) = complete graph Km
m−i m−1 0
brick polytope Ω(Xm) = conv i ∈ [m] = ,
i−1 0 m−1

59. BRICK POLYTOPE
Xm = network with two levels and m commutators
graph of flips G(Xm) = complete graph Km
m−i m−1 0
brick 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 & ASSOCIAHEDRA
triangulation 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 & ASSOCIAHEDRA
for x ∈ {a, b}n−2, define a reduced alternating network Nx and a polygon Px
5 5 5
4 a 4 a 4 a
3 a 3 a 3 b
2 a 2 b 2 a
1 1 1
2 3 4 2 3 2 4
1 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 & ASSOCIAHEDRA
For 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 & PERMUTAHEDRA
reduced network = network with n levels and n commutators
2
it supports only one pseudoline arrangement
duplicated 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 & PERMUTAHEDRA
Any pseudoline arrangement supported by Π has one contact and one crossing among
each 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
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67. THANK YOU