Automedian sets of permutations Sylvie Hamel Dpartement - - PowerPoint PPT Presentation

Automedian sets of permutations

Sylvie Hamel

Département d’informatique et de recherche opérationnelle (DIRO), Université de Montréal, Québec, Canada

Permutation Patterns 2017 June 26-30 2017, Reykjavík, Iceland

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Parti communiste du Québec Parti démocratie chrétienne du Québec Parti marxiste-léniniste du Québec

√

1 2 3 4 5 6 7 8 9

10 11

Quebec political parties :

Introduction Problem Definition Generalized problem and Conclusion Properties of medians and Automedian sets

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Kemeny consensus :

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Introduction Problem Definition Generalized problem and Conclusion Properties of medians and Automedian sets

Problem Definition Introduction

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The Kendall- distance:

Maurice Kendall

Counts the number of order disagreements between pairs of elements in two permutations i.e

The Kendall- distance is equivalent to the “bubble-sort” distance i.e. the number of transpositions needed to transform one permutation into the other one. We have

τ

dKT (π, ı) = inv(π)

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τ

Generalized problem and Conclusion

dKT (π, σ) = #{(i, j)|i < j and [(π−1

i

< π−1

j

and σ−1

i

> σ−1

j )

• r (π−1

i

> π−1

j

and σ−1

i

< σ−1

j )]}

Properties of medians and Automedian sets

Problem Definition Introduction

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Example:

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Generalized problem and Conclusion

σ π = [1, 4, 2, 5, 3]

= [3, 4, 1, 2, 5]

dKT (π, σ) =

1 1 2 2 3 3 4 4 5 5 1 + 1 + 2 2 3 3 1 + 4 4 5 5 3 3 4 4 1 + 1 5 5 4 4 5 5 + = 5

Properties of medians and Automedian sets

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π The Kendall- distance between a permutation and a set of permutations : τ A = {π1, π2, . . . , πm}

dKT (π, A) =

m

X

i=1

dKT (π, πi)

Our problem: Given a set of permutations , we want to find a permutation such that

π∗

m

A ⊆ Sn

dKT (π∗, A) ≤ dKT (π, A), ∀π ∈ Sn

Problem Definition Introduction Generalized problem and Conclusion Properties of medians and Automedian sets

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What has been done?

PP 2017 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Finding a median of a set of m permutations using the Kendall- distance

τ

Dwork et al., NP-complete for

m ≥ 4

2001

Biedl et al., first approx. algorithm + correction of Dwork et al. proof

2005 2007 Kenyon-Mathieu et Schudy, PTAS result 2008 Ailon et al., randomized algo. approx. factor 11/7

vanZuylen et al., deterministic algo. approx. factor 8/5

2009 2010 Karpinski et Schudy, fixed-parameter algorithm 2011 2013

Betzler et al., fixed-parameter algorithm Nishimura et Simjour, fixed-parameter algorithm Simjour, fixed-parameter algorithm Blin et al, space reduction Betzler et al., space reduction

2014 2016 Milosz et Hamel, space reduction

Problem Definition Introduction Generalized problem and Conclusion Properties of medians and Automedian sets

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Back to our problem:

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Given a set of permutations , we want to find a permutation such that

π∗

m

A ⊆ Sn

dKT (π∗, A) ≤ dKT (π, A), ∀π ∈ Sn

This median is not always unique

Problem Definition Introduction Generalized problem and Conclusion Properties of medians and Automedian sets

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m\n 8 10 12 14 15 20 25 30 3 2.1 3.0 3.7 4.8 5.6 12.2 23.1 61.4 4 60.6 331.4 1321.4 7551.4 14253.8

• 5

2.2 2.9 3.6 5.2 6.2 12.9 29.1 49.2 6 31.3 90.6 345.1 1506.2 1614.9

• 10

13.0 36.8 88.8 201.9 315.6 2947.9

• 15

1.7 2.2 2.8 3.5 3.8 6.3 12.3

• 20

6.3 11.4 22.2 39.8 55.5 256.7

• 25

1.6 1.9 2.3 2.6 2.9 4.6 7.6

• Average number of permutations in M(A) for uniformly distributed random sets A of m

permutations of length n. Statistics generated over 100 to 1000 instances.

Problem Definition Introduction Generalized problem and Conclusion Properties of medians and Automedian sets

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Reformulation of our problem:

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Given a set of permutations , we want to find the set of all the permutations satisfying

π∗

m

A ⊆ Sn

dKT (π∗, A) ≤ dKT (π, A), ∀π ∈ Sn

M(A)

Problem Definition Introduction Generalized problem and Conclusion Properties of medians and Automedian sets

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Properties of :

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A = {[7, 8, 2, 3, 6, 1, 5, 4], [3, 5, 1, 7, 8, 6, 2, 4], [5, 8, 3, 4, 1, 2, 7, 6]}

M(A)

Properties of medians and Automedian sets Generalized problem and Conclusion Introduction Problem Definition Space reduction

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Properties of :

M(A)

Properties of medians and Automedian sets Generalized problem and Conclusion Introduction Problem Definition Space reduction

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Let us define the following left group action: π · A We can show that is a group morphism i.e. that M π · M(A) = M(πA) πA = {π σ | σ 2 A} Sn × P(Sn) − → P(Sn)

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Automedian sets:

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Definition 1: A permutation will be called

• decomposable if , .

a i > a ⇐ ⇒ πi > a ∀i ∈ {1, 2, . . . , n}

π ∈ Sn Example: Let then is 3;4-decomposable

π = [3, 2, 1, 4, 5]

π

Definition 1’: A set of permutations is -decomposable if all

• f its permutations are -decomposable.

a a

Introduction Generalized problem and Conclusion Problem Definition Properties of medians and Automedian sets

Definition 1’’: A permutation or set will be called indecomposable if

it is not -decomposable for any .

a

a ∈ {1, 2, . . . , n − 1}

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Automedian sets:

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Introduction Generalized problem and Conclusion Problem Definition Properties of medians and Automedian sets

Definition 2: Let be an -decomposable set and let be any permutation. Then the set is called

• separable.

If is not separable for any , it is called inseparable. a a σ ∈ Sn A ⊆ Sn σA A ⊆ Sn a ∈ {1, . . . , n − 1} Example: is a 3-decomposable set. Let , then A = {[3, 2, 1, 5, 4], [3, 1, 2, 4, 5], [1, 2, 3, 5, 4]} σ = [4, 2, 1, 3, 5] σA = {[1, 2, 4, 5, 3], [1, 4, 2, 3, 5], [4, 2, 1, 5, 3]} which is 3-separable. ,

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Automedian sets:

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Definition 3: Let and be two permutations

• f length and , respectively. The direct sum of

and , denoted , is defined as

π ∈ Sk σ ∈ S` `

k

π

σ

π ⊕ σ Example: Let and π = [3, 2, 1, 4, 5] σ = [1, 3, 2] π ⊕ σ = π1π2 . . . πk(σ1 + k)(σ2 + k) . . . (σ` + k) then π ⊕ σ = [3, 2, 1, 4, 5, 6, 8, 7]

Introduction Generalized problem and Conclusion Problem Definition Properties of medians and Automedian sets

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Automedian sets:

Definition 4: Let and be two set of

• permutations. The direct sum of and , denoted ,

is defined as A ⊂ Sk B ⊂ S` A B A ⊕ B A ⊕ B = {π ⊕ σ | π ∈ A and σ ∈ B} A ⊕ B is -decomposable k

Introduction Generalized problem and Conclusion Problem Definition

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Automedian sets:

Example: Let and A = {[1, 3, 2], [3, 1, 2]} B = {[2, 1, 4, 3], [2, 3, 1, 4] [2, 4, 3, 1]} then A ⊕ B = {[1, 3, 2, 5, 4, 7, 6], [1, 3, 2, 5, 6, 4, 7], [1, 3, 2, 5, 7, 6, 4], [3, 1, 2, 5, 4, 7, 6], [3, 1, 2, 5, 6, 4, 7], [3, 1, 2, 5, 7, 6, 4]}

Introduction Generalized problem and Conclusion Problem Definition

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Automedian sets:

Theorem 1: M(A ⊕ B) = M(A) ⊕ M(B) Theorem 2: and A = M(A) B = M(B) ⇐ ⇒ A ⊕ B = M(A ⊕ B) Theorem 3: If is an -decomposible automedian set, then automedian sets such that . C ⊂ Sn a ∃ A ⊂ Sa, B ⊂ Sn−a C = A ⊕ B

Introduction Generalized problem and Conclusion Problem Definition

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Counting automedian sets:

Definition: AMn = {A = M(A) | A ⊆ Sn} |AMn| = |In| +

n−1

X

i=1

✓n i ◆ × |Ii| × |AMn−i| |In| : 1, 1, 3, 27, . . . |AMn| : 1, 3, 15, 117, . . .

Introduction Generalized problem and Conclusion Problem Definition

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Properties of medians and Automedian sets

Definition: In = {A = M(A) | A ⊆ Sn and A inseparable}

What’s left to do:

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A lot!

• Complexity in the case where m=3
• Completely solve the automedian case
• Investigate the shuffle of sets of permutations
• Do the same kind of investigation for the problem of

finding a median of other kind of combinatorial objects

Generalized problem and Conclusion Introduction Problem Definition

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Properties of medians and Automedian sets

Election issues :

health taxes infrastructures environment education independence 20 / 22

Generalized problem and Conclusion Introduction Problem Definition

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1 2 3 1 2 3 4 5 1 2 3 1 2 3 4 1 2 3 4

Generalized Kemeny consensus :

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* B. Brancotte, B.Yang, G.Blin, S. Cohen-Boulakia,

• A. Denise and S.Hamel, Rank aggregation with ties:

Experiments and Analysis, Proceedings of the VLDB (Very Large Data Bases) Endowment. 8(11): 1202-1213, 2015

Generalized problem and Conclusion Introduction Problem Definition

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Properties of medians and Automedian sets

Collaborators:

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Automedian:

Charles Desharnais Robin Milosz

Generalized problem:

Bryan Brancotte Alain Denise Sarah Cohen-Boulakia

+ all the summer research trainees

Guillaume Blin

Generalized problem and Conclusion Introduction Problem Definition

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Properties of medians and Automedian sets

Questions

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Generalized problem and Conclusion Introduction Problem Definition

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Properties of medians and Automedian sets

Spurningar

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