Thesis Defence for Doctor of Information Science

Studies on Permutation Set ManipulationBased on Decision Diagrams

January 23, 2017

Large-Scale Knowledge Processing LaboratoryYuma Inoue

Doctoral(Thesis(Defense(Presentation(

Permutations1 Bijection(π: [n]→ [n] on(a(set([n] = {1, 2, …, n}.

1 Notation:((((((((((((((((((((((((((((((((((((((((((((or

1 Ordering(of(items:(ex.((2, 3, 1, 4) for(a(set(S = {1, 2, 3, 4}.

1 Permutations(appear(in(several(applications:

matching,(sorting,(scheduling,(ordering,(encoding,(…

2

1

1

2

2

3

3

4

453

6

2

14


Permutation Sets

3

1 A(subset(Π� Sn of(all(permutations(on([n].1 Collection(of(solutions

53

6

2

14 { (1, 3, 4, 2, 5, 6),

(1, 3, 4, 2, 6, 5),(1, 4, 3, 2, 5, 6),(1, 4, 3, 2, 6, 5) }

1 Useful(for…

1 extraction(with(criteria,

1 modification,

1 analysis,(etc.


Permutation Sets

4


53

6

2

14 { (1, 3, 4, 2, 5, 6),

(1, 3, 4, 2, 6, 5),(1, 4, 3, 2, 5, 6),(1, 4, 3, 2, 6, 5) }

1 Useful(for…


1 modification,

1 analysis,(etc.

Extract(π s.t. π(3) = 3


Permutation Sets

5


53

6

2

14 { (1, 3, 4, 2, 5, 6),

(1, 3, 4, 2, 6, 5),(1, 4, 3, 2, 5, 6),(1, 4, 3, 2, 6, 5) }

1 Useful(for…


1 modification,

1 analysis,(etc.

Add(edge((3,4)


Permutation Sets

6


53

6

2

14 { (1, 3, 4, 2, 5, 6),

(1, 3, 4, 2, 6, 5),(1, 4, 3, 2, 5, 6),(1, 4, 3, 2, 6, 5) }

1 Useful(for…


1 modification,

1 analysis,(etc.

How(many(π’s(satisfy(π(5) > π(6)?


Permutation Sets

7


53

6

2

14 { (1, 3, 4, 2, 5, 6),

(1, 3, 4, 2, 6, 5),(1, 4, 3, 2, 5, 6),(1, 4, 3, 2, 6, 5) }

1 The(number(of(permutations:(n!→ Store(permutations(into(compressed(data(structure

with(operations.


Data structure for Permutation SetsπDD([Minato11]1 Compress(any(permutation(set(with(manipulations.

1 Union,(Intersection,

1 Cartesian(product((((((((((((((((((((((((((((((((((((,

1 Extract(with(criteria((ex.(π(i) = j),(etc.1 Unknown(theoretical(size(bound(in(general.

8

10

τ1,2

τ1,3

τ1,4

{ (2, 1, 3, 4) = τ1,2,(2, 4, 3, 1) = τ1,4 τ1,3,(3, 2, 4, 1) = τ1,4 τ1,2 }

Main(target(of(this(thesis


Topics in the Thesis

9

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

Application Problems

Data structures

πDD

ZDDderive

Chapter(2


Application ProblemsTopics in the Thesis

10

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

Chapter(3

πDD

ZDD

Data structures [Reversible(Computation(15]



11

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

[IEICE Trans.14]

πDD

ZDD

Data structures



12

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.1

ZDD

Data structures



13

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Chapter(4

[ISAAC14]

ZDD

Data structures



14

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.5

?

ZDD

Data structures



15

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

ZDD

Subgraph

Enumeration

Chapter(5 improves

Data structures


Degree Requirements�Y.(Inoue,(T.(Toda,(and(M.(Minato."Implicit(Generation(of(Pattern1Avoiding(Permutations(by(Using(Permutation(Decision(Diagrams."IEICE(TRANSACTIONS(on(Fundamentals(of(Electronics,(Communications(and(Computer(Sciences,(Vol.E971A,(No.6,(pp.117111179,(June(2014.

�Y.(Inoue and(S.(Minato."An(Efficient(Method(for(Indexing(All(Topological(Orders(of(a(Directed(Graph."Algorithms(and(Computation(1 25th(International(Symposium((ISAAC),pp.1031114(2014.

�Y.(Inoue and(S.(Minato."Improved(Algorithms(for(Debugging(Problems(on(Erroneous(Reversible(Circuits."Reversible(Computation(1 7th(International(Conference,((RC),(pp.1861199,(2015.

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Section 4.4

Section 3.1

Section 4.3


Degree Requirements�Y.(Inoue,(T.(Toda,(and(M.(Minato."Implicit(Generation(of(Pattern1Avoiding(Permutations(by(Using(Permutation(Decision(Diagrams."IEICE(TRANSACTIONS(on(Fundamentals(of(Electronics,(Communications(and(Computer(Sciences,(Vol.E971A,(No.6,(pp.117111179,(June(2014.

�Y.(Inoue and(S.(Minato."An(Efficient(Method(for(Indexing(All(Topological(Orders(of(a(Directed(Graph."Algorithms(and(Computation(1 25th(International(Symposium((ISAAC),pp.1031114(2014.

�Y.(Inoue and(S.(Minato."Improved(Algorithms(for(Debugging(Problems(on(Erroneous(Reversible(Circuits."Reversible(Computation(1 7th(International(Conference,((RC),(pp.1861199,(2015.

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Section 4.4

Section 4.3

Section 3.1


Application ProblemsNext Topic

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Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

ZDD

Subgraph

Enumeration

Data structures


ZDDs [Minato93]�Data(structure(for(a(family(of(sets.

�Compressed(binary(decision(tree(by:

1 delete(nodes(whose(11edge(pointing(to(01terminal,1 merge(the(equivalent(nodes.

�ZDD(supports(several(operations:

1 union,

1 intersection,(etc.

�ZDD(operation(timedepends(only(on

the(size(of(ZDD. 01101000

cccc

b b

a10

10

c

b b

a 10

{ {a, b}, {a, c}, {b, c} }

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πDDs [Minato11]�Data(structure(for(a(set(of(permutations.

�Transposition(τi,j:(�Any(permutation(can(be(decomposed(into(sequence(of τi,j.Ex.((2, 4, 3, 1) ← (4, 2, 3, 1) ← (1, 2, 3, 4)

�Transpositions(are(assigned(to(ZDD(nodes.

�πDDs(have(several(operations:1 Cartesian(product,

1 extraction,(etc.

1 Runtime(dependson(the(size(of(πDDs.

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τ1,2

τ1,3

τ1,4

{ (2, 1, 3, 4) = τ1,2,(2, 4, 3, 1) = τ1,4 τ1,2,(3, 2, 4, 1) = τ1,4 τ1,3 }

τ1,2 τ1,4

20



21

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

ZDD

Subgraph

Enumeration

Data structures


x1

x2

x3 y3

y2

y1

0

0

1

0

1

1

1

1

1

1

1

0

g1 g2 g3

01

23

4

567

x1 x2 x3 y3y2y1

0 0 00 00 00

111 10 00

01

11 11

111

0 0 10 0 0

0 010 1 1

0 11011

1 110 01

G = g1g2g3f

⇡f

: controls : targets

Debugging Erroneous Reversible CircuitsProblemInput:(an(objective(function,(an(erroneous(reversible(circuit

Output:(error(positions(and(ways(to(fix(them

Contributions:1 For(single1error(model:(propose(an(algorithm((without(πDDs)(faster(than(the(existing(methods([Wille+09]([Tague+13].

1 For(multiple1error(model:(propose(the(first(debugging(algorithm(by(utilizing(πDDs.

1 Reversible(Circuit:(specify(a(permutation(on({0, 1, …, 2n-1}j1 described(as(a(cascade(of(gates.

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Cycle-type Partition of Permutation SetsProblemInput:(a(permutation(set

Output:(partition(with(respect(to(cycle1type(equivalence

Contributions:1 A(proposed(πDD1based(algorithm(runs(fast(and(uses(less(memory(than(a(previous(πDD1based(approach([Yamada+12].

1 A(permutation(is(decomposed(into(cycles.

1 Cycle1type(equivalence(=(the(same(topology(of(cycles.

1

2

3

4

5

6

1

2

3

4

5

6

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Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

ZDD

Data structures


Rotation-based πDDs (Rot-πDDs)�Modified(version(of(πDD�Left1rotation(ρi,j:(

�Any(permutation(can(be(decomposed(into(sequence(of ρi,j.

Ex.((3, 2, 4, 1) ← (2, 3, 4, 1) ← (1, 2, 3, 4)

�Left1rotations(are(assigned(to(ZDD(nodes.

�Operations(on(πDDs(are(also(available.

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ρ1,2

ρ1,3

ρ1,4

{ (2, 1, 3, 4) = ρ1,2,(2, 4, 3, 1) = ρ1,4 ρ1,3,(3, 2, 4, 1) = ρ1,4 ρ1,2 }

ρ1,2 ρ1,4

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Enumeration of Topological Orders

1 Topological(order:(a(sequence(of(vertices(such(that(the(order(

must(not(contradict(all(edges.

Contributions:1 Give(dynamic(programming(algorithm(for(Rot1πDD(construction.1 Provide(a(Rot1πDD(operation(to(handle(dynamic(edge(addition.1 Show(Rot1πDDs(are(theoretically(preferable(to(πDDs.

Input:(a(directed(graph

Output:(all(topological(orders(in(the(graph

Problem

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1 4

53 621 4

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1 Dynamic(programming(for(counting:

1 Recursively(compute(the(number(of(topological(orders

for(each(induced(subgraph(of(graph(G:(dpS = Σv�S dpS�{v}.

1 Reduce(search(space:(n! → 2n (n =|V|)

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6

4

1 2

dp{3,4,5,6} = 2

dp{2,3,4,5,6} = 4

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Key idea for Rot-πDD construction algorithm

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4

1 2dp{2,4,5,6} = 2 53

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1 2


1 Permutation(representation(of(search(states:

1 k1suffix:(reverse(of(used(vertices,((n-k)1prefix:(unused(vertices1 (n-k)1prefix(corresponds(to(an(inducing(vertex(set.

1 transposition(τx,y or(left1rotation(ρx,y:use(the(x1th unused(vertex(as(the((n-y)1th vertex.

(2, 3, 4, 5, 6, 1)

ρ2,5

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53

6

4

1 2

53

6

4

1 2

(2, 4, 5, 6, 3, 1)


1 transpositions:(two(different(k1prefixes(represent(the(same(state.1 left1rotation:(k1prefix(is(in(the(increasing(order.

← left1rotations(keep(relative(orders(of(prefix.

(6, 2, 3, 4, 5, 1)

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53

6

4

1 2

(6, 5, 3, 4, 2, 1)

(6, 5, 4, 3, 2, 1)

(6, 2, 5, 4, 3, 1)

(6, 4, 5, 2, 3, 1)

(2, 3, 4, 5, 6, 1)

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6

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1 2

(3, 4, 5, 6, 2, 1)

(4, 5, 6, 3, 2, 1)

(2, 4, 5, 6, 3, 1)

(4, 5, 6, 2, 3, 1)

× ○

ρ2,5ρ1,5

ρ1,4 ρ1,4

τ3,5τ2,5

τ3,4 τ2,4


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(2, 3, 4, 5, 6, 1)

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1 2

(3, 4, 5, 6, 2, 1) (2, 4, 5, 6, 3, 1)

ρ2,5ρ1,5

ρ1,4 ρ1,4

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1 2Rot1πDD(for

1 transpositions:(two(different(k1prefixes(represent(the(same(state.1 left1rotation:(k1prefix(is(in(the(increasing(order.

← left1rotations(keep(relative(orders(of(prefix.


a

b c

d e

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3 6

2 5

Enumeration of Euler Paths

1 Euler(path:(a(walk(passing(through every(edge(exactly(once.

Contributions:1 Provide(construction(algorithm(of(Rot1πDD(representing(all(Euler(paths(based(on(dynamic(programming(for(counting.

1 Show(Rot1πDDs(are(theoretically(preferable(to(πDDs.

Input:(an(undirected(graph

Output:(all(Euler(paths(in(the(graph

Problem

b→ d → a → b → c → e → d3 2 1 4 6 5

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Enumeration of Pattern-Avoiding Permutations

Contributions:1 Provide(a(Rot1πDD(construction(algorithm,(which(is(faster(than(a(commonly(used(software((=(PermLab [Albert12]).

1 Reveal(the(advantage(of(Rot1πDDs(against(πDDs.

Input:(permutation(pattern(s),(an(integer(nOutput:(all(the(permutations(on([n] avoiding(the(pattern(s).

Problem

1 Text(perm(τ avoids(pattern(perm(π�no(subsequence(of(τ has(the(same(relative(order(to(π.Ex.(Text((4,(2,(1,(3),(Pattern((3,(1,(2)

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33

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.5

?

ZDD

Data structures


Application ProblemsChoice of Permutation Decision Diagrams

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τ2,3

(2, 3, 4, 5, 6, 1)

ρ2,553

6

4

1 2

53

6

4

1 2

(2, 4, 5, 6, 3, 1)

Conduct(experiments(to(investigate(permutation(parameters(

affecting(the(performance(of(decision(diagrams.

1 Focus(on(disorder(measures(as(permutation(parameters:(

Inv,%Dis,%Exc,%Max,%Res,(and(Runs.1 Rot1πDDs(are(preferable(to(πDDs(except(Exc.



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Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

ZDD

Subgraph

Enumeration

Data structures


Enumeration of Subgraphs

1 A(ZDD(can(represent(subgraphs(of(a(graph.

1 A(variable(order(of(ZDDs(dramatically(affects(the(size(of(ZDDs.

Input:(an(undirected(graph(and(constraints

Output:(ZDD(for(all(the(subgraph(satisfying(the(constraints

Problem

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Enumeration of SubgraphsKey observation to minimize ZDDs1 The(size(of(ZDDs(is(related(to(the(path1width(of(a(graph.

1 Propose(a(method(computing(an(edge(order(with(small(path1width.

1 Improves(the(size(of(ZDDs(for(benchmark.

Contributions:


Conclusions

1 Utilize(πDDs(for(two(problems.1 Propose(Rot1πDDs(and(utilize(it(for(three(problems.

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Apply%decision%diagrams%to%permutation%problems.

1 Properties(of(individual(problems:(swaps(and(relative(orders.

1 Permutation(parameters:(disorder(measures.

Analyze%the%performance%of%decision%diagrams%experimentally%and%theoretically.


Open Problems�Optimization(schemes(on(permutation(decision(diagrams.

�Reveal(theoretical(relations(between(permutation(decision(diagrams(and(permutation(parameters.

�Generalized(framework(for(permutation(decision(diagram(based(on(generators(of(permutation(groups.

1 Transpositions(and(left1rotations(are(generators of

a(symmetric(group.

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Thesis Defence for Doctor of Information Science

Science

Transcript of Thesis Defence for Doctor of Information Science