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.
Time Machine session @ ICME 2012 - DTW's New YouthXavier Anguera
This presentation are the slides I gave at the Time Machine Expert session of ICME 2012. It talks about the renewal of Dynamic Time Warping (DTW) as a feasible algorithm for some of today's applications.
Multimodal pattern matching algorithms and applicationsXavier Anguera
In this presentation I focus on 3 projects I have been working in the last year. The first one is a novel pattern matching algorithm, based on the well known Dynamic Time Warping. The presented algorithm can be used to find real-valued subsequences within a longer sequence, without prior knowledge of their start-end points. I have applied the algorithm for the task of acoustic matching, for which I will show some preliminary results. Then I will continue to explain a second DTW-based algorithm, this one being able do an online of two musical pieces. One of the music pieces can be input life or be retrieved from an audio file, while the second one is extracted from an online music video. The online alignment allows for the music video to be played in total synchrony with the corresponding ambient/recorded audio. Finally, I will talk about video copy detection, which is the task of finding video duplicate segments within a big database. I will explain our multimodal approach, based on audio-visual change-based features.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Nelly Litvak – Asymptotic behaviour of ranking algorithms in directed random ...Yandex
There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.
In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.
Time Machine session @ ICME 2012 - DTW's New YouthXavier Anguera
This presentation are the slides I gave at the Time Machine Expert session of ICME 2012. It talks about the renewal of Dynamic Time Warping (DTW) as a feasible algorithm for some of today's applications.
Multimodal pattern matching algorithms and applicationsXavier Anguera
In this presentation I focus on 3 projects I have been working in the last year. The first one is a novel pattern matching algorithm, based on the well known Dynamic Time Warping. The presented algorithm can be used to find real-valued subsequences within a longer sequence, without prior knowledge of their start-end points. I have applied the algorithm for the task of acoustic matching, for which I will show some preliminary results. Then I will continue to explain a second DTW-based algorithm, this one being able do an online of two musical pieces. One of the music pieces can be input life or be retrieved from an audio file, while the second one is extracted from an online music video. The online alignment allows for the music video to be played in total synchrony with the corresponding ambient/recorded audio. Finally, I will talk about video copy detection, which is the task of finding video duplicate segments within a big database. I will explain our multimodal approach, based on audio-visual change-based features.
Computational Information Geometry: A quick review (ICMS)Frank Nielsen
From the workshop
Computational information geometry for image and signal processing
Sep 21, 2015 - Sep 25, 2015
ICMS, 15 South College Street, Edinburgh
http://www.icms.org.uk/workshop.php?id=343
Nelly Litvak – Asymptotic behaviour of ranking algorithms in directed random ...Yandex
There is a vast empirical research on the behaviour of ranking algorithms, e.g. Google PageRank, in scale-free networks. In this talk, we address this problem by analytical probabilistic methods. In particular, it is well-known that the PageRank in scale-free networks follows a power law with the same exponent as in-degree. Recent probabilistic analysis has provided an explanation for this phenomenon by obtaining a natural approximation for PageRank based on stochastic fixed-point equations. For these equations, explicit solutions can be constructed on weighted branching trees, and their tail behavior can be described in great detail.
In this talk we present a model for generating directed random graphs with prescribed degree distributions where we can prove that the PageRank of a randomly chosen node does indeed converge to the solution of the corresponding fixed-point equation as the number of nodes in the graph grows to infinity. The proof of this result is based on classical random graph coupling techniques combined with the now extensive literature on the behavior of branching recursions on trees.
Tools for Modeling and Analysis of Non-manifold ShapesDavid Canino
My Official PhD thesis is available at http://www.disi.unige.it/dottorato/THESES/2012-01-CaninoD.pdf
They are slides, that I presented on May 7, 2012, while defending my PhD Thesis in Computer Science under the supervision of Professor Leila De Floriani at the DIBRIS Department (Department of Bioengineering, Computer Science, and Systems Engineering) in Genova, Italy.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Tools for Modeling and Analysis of Non-manifold ShapesDavid Canino
My Official PhD thesis is available at http://www.disi.unige.it/dottorato/THESES/2012-01-CaninoD.pdf
They are slides, that I presented on May 7, 2012, while defending my PhD Thesis in Computer Science under the supervision of Professor Leila De Floriani at the DIBRIS Department (Department of Bioengineering, Computer Science, and Systems Engineering) in Genova, Italy.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
Generalized CDT as a scaling limit of planar mapsTimothy Budd
Generalized causal dynamical triangulations (generalized CDT) is a model of two-dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur. After identifying the model as a scaling limit of random quadrangulations, I will show how it can be solved using a bijection between quadrangulations and trees. Another bijection relating quadrangulations to planar maps allows us to interpret generalized CDT as a scaling limit or random planar maps with a restriction on the number of faces. Finally I will show how this interpretation clarifies certain mysterious identities in generalized CDT amplitudes. (This talk is largely based on arXiv:1302.1763.)
On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph f...Editor IJCATR
The main purpose of this paper is to study the zero-divisor graph for direct product of finite commutative rings. In our
present investigation we discuss the zero-divisor graphs for the following direct products: direct product of the ring of integers under
addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p2 for a prime number p,
direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and
multiplication modulo 2p for an odd prime number p and direct product of the ring of integers under addition and multiplication
modulo p and the ring of integers under addition and multiplication modulo p2 – 2 for that odd prime p for which p2 – 2 is a prime
number. The aim of this paper is to give some new ideas about the neighborhood, the neighborhood number and the adjacency matrix
corresponding to zero-divisor graphs for the above mentioned direct products. Finally, we prove some results of annihilators on zerodivisor
graph for direct product of A and B for any two commutative rings A and B with unity
In this article we consider macrocanonical models for texture synthesis. In these models samples are generated given an input texture image and a set of features which should be matched in expectation. It is known that if the images are quantized, macrocanonical models are given by Gibbs measures, using the maximum entropy principle. We study conditions under which this result extends to real-valued images. If these conditions hold, finding a macrocanonical model amounts to minimizing a convex function and sampling from an associated Gibbs measure. We analyze an algorithm which alternates between sampling and minimizing. We present experiments with neural network features and study the drawbacks and advantages of using this sampling scheme.
Challenges in predicting weather and climate extremesIC3Climate
Presentation from the Kick-off Meeting "Seasonal to Decadal Forecast towards Climate Services: Joint Kickoff Meetings" for ECOMS, EUPORIAS, NACLIM and SPECS FP7 projects
A function f is called a graceful labelling of a graph G with q edges if f is an injection
from the vertices of G to the set {0, 1, 2, . . . , q} such that, when each edge xy is assigned the
label |f(x) − f(y)| , the resulting edge labels are distinct. A graph G is said to be one modulo
N graceful (where N is a positive integer) if there is a function φ from the vertex set of G to
{0, 1,N, (N + 1), 2N, (2N + 1), . . . ,N(q − 1),N(q − 1) + 1} in such a way that (i) φ is 1 − 1 (ii)
φ induces a bijection φ_ from the edge set of G to {1,N + 1, 2N + 1, . . . ,N(q − 1) + 1} where
φ_(uv)=|φ(u) − φ(v)| . In this paper we prove that the every regular bamboo tree and coconut tree
are one modulo N graceful for all positive integers N .
An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles graphhoc
In 1991, Gnanajothi [4] proved that the path graph n
P with n vertex and n −1edge is odd graceful, and
the cycle graph Cm with m vertex and m edges is odd graceful if and only if m even, she proved the
cycle graph is not graceful if m odd. In this paper, firstly, we studied the graphCm∪Pn when m = 4, 6,8,10
and then we proved that the graphCm∪Pn
is odd graceful if m is even. Finally, we described an
algorithm to label the vertices and the edges of the vertex set ( ) m n
V C ∪P and the edge set ( ) m n
E C ∪P .
Continuum Modeling and Control of Large Nonuniform NetworksYang Zhang
Presented at The 49th Annual Allerton Conference on Communication, Control, and Computing, 2011
Abstract—Recent research has shown that some Markov chains modeling networks converge to continuum limits, which are solutions of partial differential equations (PDEs), as the number of the network nodes approaches infinity. Hence we can approximate such large networks by PDEs. However, the previous results were limited to uniform immobile networks with a fixed transmission rule. In this paper we first extend the analysis to uniform networks with more general transmission rules. Then through location transformations we derive the continuum limits of nonuniform and possibly mobile networks. Finally, by comparing the continuum limits of corresponding nonuniform and uniform networks, we develop a method to control the transmissions in nonuniform and mobile networks so that the continuum limit is invariant under node locations, and hence mobility. This enables nonuniform and mobile networks to maintain stable global characteristics in the presence of varying node locations.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Jean Cardinal (Computer Science Department, Université Libre de Bruxelles)
Sorting and a Tale of Two Polytopes
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Mireille Bousquet-Mélou (LABRI, CNRS)
The Number of Inversions After n Adjacent Transpositions
Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Olivier Hudry (INFRES-MIC2 Télécom ParisTech)
A Branch and Bound Algorithm to Compute a Median Permutation
Algorithms & Permutations 2012, Paris.
http://igm.univ-mlv.fr/AlgoB/algoperm2012/
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
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+
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
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+
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
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
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
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