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,(…

Permutation Sets

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

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

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

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

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

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.

{ (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

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

Application Problems

Data structures

ZDDderive

Chapter(2

Application ProblemsTopics in the Thesis

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

Chapter(3

Data structures [Reversible(Computation(15]

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

[IEICE Trans.14]

Data structures

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.1

Data structures

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Chapter(4

[ISAAC14]

Data structures

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.5

Data structures

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

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.

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.

Section 4.4

Section 4.3

Section 3.1

Application ProblemsNext Topic

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

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

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

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.

{ (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

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Subgraph

Enumeration

Data structures

g1 g2 g3

x1 x2 x3 y3y2y1

0 0 00 00 00

111 10 00

0 0 10 0 0

0 010 1 1

0 11011

1 110 01

G = g1g2g3f

: 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.

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.

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

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.

{ (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

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

53 621 4

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|)

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

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

Key idea for Rot-πDD construction algorithm

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

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)

(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)

(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)

(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

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

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

ρ2,5ρ1,5

ρ1,4 ρ1,4

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.

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

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)

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

Section(4.5

Data structures

Application ProblemsChoice of Permutation Decision Diagrams

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

ρ2,553

(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.

Reversible(Circuit

Debugging

Cycle1type

Partition

Euler(Path

Enumeration

Topological(Order

Enumeration

Pattern1avoiding

Perm.(Enumeration

πDD Rot-πDD

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

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.

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.

Thesis Defence for Doctor of Information Science

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Transcript of Thesis Defence for Doctor of Information Science

Msc Thesis

Laiki Ki8ara Thesis

Mars2030 Thesis Booklet

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PhD thesis defence presentation

Hamza thesis 12.7.15

THESIS 19MAR2012

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phd thesis presentation

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Flag-Transitive Symmetric Designs - UNAMeugenia/papers/thesis.pdfFlag-Transitive Symmetric Designs Eugenia O’Reilly Regueiro Thesis submitted for the degree of Doctor of Philosophy

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Classical and quantum dynamics of optical frequency ......Classical and quantum dynamics of optical frequency conversion By Andrew G. White A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR

HE DEFENCE INDUSTRY AS AN EXPLANATORY FACTOR OF THE GERMAN …

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