19 topological sort - ETH Zse.inf.ethz.ch/.../0001/slides/19_topological_sort_6up.pdf3 Intro. to...

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Intro. to Programming, lecture 19: Topological Sort I 1

Einführung in die ProgrammierungIntroduction to Programming

Prof. Dr. Bertrand MeyerOctober 2006 – February 2007

Chair of Software Engineering

Lecture 19: Topological sortPart 1: Problem and math basis


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by Caran d’Ache


“Topological sort”

From a given partial order, produce a compatible total order


The problem

Partial order: ordering constraints between elements of a set, e.g.“Remove the dishes before discussing politics”“Walk to Üetliberg before lunch”“Take your medicine before lunch”“Finish lunch before removing dishes”

Total order: sequence including all elements of setCompatible: the sequence respects all ordering constraints

Üetliberg, Medicine, Lunch, Dishes, Politics : OKMedicine, Üetliberg, Lunch, Dishes, Politics : OKPolitics, Medicine, Lunch, Dishes, Üetliberg : not OK

From a given partial order,produce a compatible total order


Why we are doing this!

Very common problem in many different areasInteresting, efficient, non-trivial algorithmIllustration of many algorithmic techniquesIllustration of data structures, complexity (big-Oh

notation), and other topics of last lectureIllustration of software engineering techniques: from

algorithm to component with useful APIOpportunity to learn or rediscover important

mathematical concepts: binary relations (order relations in particular) and their properties

It’s just beautiful!Today: problem and math basisNext time: detailed algorithm and component

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Reading assignment for next Monday

Touch of Class, chapter on topological sort: 17


Topological sort: example uses

From a dictionary, produce a list of definitions such that no word occurs prior to its definition

Produce a complete schedule for carrying out a set of tasks, subject to ordering constraints

(This is done for schedulingmaintenance tasks for industrial tasks, often with thousands of constraints)

Produce a version of a class with the features reordered so thatno feature call appears before the feature’s declaration


Rectangles with overlap constraints

BD

A

CE

Constraints: [B, A], [D, A], [A, C], [B, D], [D, C]


Rectangles with overlap constraints

BD

A

CE

B D E A C

Constraints: [B, A], [D, A], [A, C], [B, D], [D, C]Possible solution:


The problem

Partial order: ordering constraints between elements of a set, e.g.“Remove the dishes before discussing politics”“Walk to Üetliberg before lunch”“Take your medicine before lunch”“Finish lunch before removing dishes”

Total order: sequence including all elements of setCompatible: the sequence respects all ordering constraints

Üetliberg, Medicine, Lunch, Dishes, Politics : OKMedicine, Üetliberg, Lunch, Dishes, Politics : OKPolitics, Medicine, Lunch, Dishes, Üetliberg : not OK

From a given partial order,produce a compatible total order


Pictured as a graph

“Remove the dishes before discussing politics”“Walk to Üetliberg before lunch”“Take your medicine before lunch”

“Finish lunch before removing dishes”

Üetliberg

MedicineLunch Dishes Politics

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Sometimes there is no solution

“Introducing recursion requires that students know about stacks”

“You must discuss abstract data typesbefore introducing stacks

“Abstract data types rely on recursion”

The constraints introduce a cycle


Overall structure (1)

Given:

Required:An enumeration of the elements, in an order compatible with the constraints

A type G

A set of elementsof type G

constraints: LIST [TUPLE [G, G ]]

topsort : LIST [G ] is...ensure

compatible (Result, constraints)

class ORDERABLE [G] feature

end

A set of constraints between these elements

elements: LIST [G ]


Some mathematical background...


Binary relation on a set

Any property that either holds or doesn’t hold between two elements of a set

On a set PERSON of persons, example relations are:mother : a mother b holds if and only if a is the

mother of bfather :

child :

sister:

sibling : (brother or sister)

Notation: a r b to express that r holds of a and b.

Note: relation names will appear in green


Example: the before relation

The set of interest:Tasks = {Politics, Lunch, Medicine, Dishes, Üetliberg}

The constraining relation:Dishes before PoliticsÜetliberg before LunchMedicine before LunchLunch before Dishes

“Remove the dishes before discussing politics”“Walk to Üetliberg before lunch”“Take your medicine before lunch”



Some special relations on a set X

universal [X ]: holds between any two elements of X

id [X ]: holds between every element of X and itself

empty [X ]: holds between no elements of X

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Relations: a more precise mathematical view

We consider a relation r on a set P as:

A set of pairs in P x P,containing all the pairs [x, y] such that x r y.

Then x r y simply means that [x, y] ∈ r

See examples on next slide


Example: the before relation

The set of interest:elements = {Politics, Lunch, Medicine, Dishes, Üetliberg}

The constraining relation:before =

{[Dishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, Dishes]}

“Remove dishes before discussing politics”“Walk to Üetliberg before lunch”“Take your medicine before lunch”



Using ordinary set operators

spouse = wife ∪ husband

sibling = sister ∪ brother ∪ id [Person]

sister ⊆ sibling

father⊆ ancestor

universal [X ] = X x X (cartesian product)empty [X ] = ∅


Possible properties of a relation

(On a set X. All definitions must hold for every a, b, c... ∈ X.)

Total*: (a ≠ b) ⇒ ((a r b) ∨ (b r a))Reflexive: a r aIrreflexive: not (a r a)Symmetric: a r b ⇒ b r aAntisymmetric: (a r b) ∧ (b r a) ⇒ a = bAsymmetric: not ((a r b) ∧ (b r a))

Transitive: (a r b) ∧ (b r c) ⇒ a r c

*Definition of “total” is specific to this discussion (there is no standard definition). The other terms are standard.


Examples (on a set of persons)

sibling

Total: (a ≠ b) ⇒ (a r b) ∨ (b r a)Reflexive: a r aIrreflexive: not (a r a)Symmetric: a r b ⇒ b r aAntisymmetric: (a r b) ∧ (b r a) ⇒ a = bAsymmetric: not ((a r b) ∧ (b r a))Transitive: (a r b) ∧ (b r c) ⇒ a r c

Reflexive, symmetric, transitive

sister Symmetric, irreflexiveReflexive, antisymmetricfamily_head

(a family_head b means a is the headof b’s family, with one head per family)

mother Asymmetric, irreflexive


Total order relation (strict)

Relation is strict total order if:TotalIrreflexiveTransitive

Example: “less than” < on natural numbers

0 < 1 0 < 2, 0 < 3, 0 < 4, ...1 < 2 1 < 3 1 < 4, ...

2 < 3 2 < 4, ......

0 1 2 3 4 5

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a))Irreflexive: not (a r a)Symmetric: a r b ⇒ b r aAsymmetric: not ((a r b) ∧ (b r a))


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Theorem

A strict order is asymmetric(total)




Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a))Irreflexive: not (a r a)Symmetric: a r b ⇒ b r aAsymmetric: not ((a r b) ∧ (b r a))


Theorem:

A strict order is asymmetrictotal


Total order relation (non-strict)

Relation is non-strict total order if:TotalReflexiveTransitiveAntisymmetric

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a))Irreflexive: not (a r a)Symmetric: a r b ⇒ b r aAntisymmetric: (a r b) ∧ (b r a) ⇒ a = bTransitive:(a r b) ∧ (b r c) ⇒ a r c

Example: “less than or equal” ≤ on natural numbers

0 ≤ 0 0 ≤ 1, 0 ≤ 2, 0 ≤ 3, 0 ≤ 4, ...1 ≤ 1 0 ≤ 2, 0 ≤ 3, 0 ≤ 4, ...

1 ≤ 2, 1 ≤ 3, 1 ≤ 4, ...2 ≤ 2 2 ≤ 3, 2 ≤ 4, ...

...

0 1 2 3 4 5




Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a))

Irreflexive: not (a r a)Symmetric: a r b ⇒ b r aAntisymmetric: (a r b) ∧ (b r a) ⇒ a = bTransitive:(a r b) ∧ (b r c) ⇒ a r c


Partial order relation (strict)

[1, 2]

1

2

b

a[0, 1]

y

x0

1 2 3

[3, 0]

c

[4, 2]

dExample: relation between points in a plane:

p q if bothxp < xqyp < yq

<

Relation is strict partial order if:TotalIrreflexiveTransitive

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a))

Irreflexive: not (a r a)Symmetric: a r b ⇒ b r aAntisymmetric: (a r b) ∧ (b r a) ⇒ a = bTransitive:(a r b) ∧ (b r c) ⇒ a r c


Theorems

A strict order is asymmetric(total)partial

A total order is a partial order

(“partial” order really means possibly partial)

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Example partial order

Here the following hold:

c a

No link between a and c, b and c:

a c nore.g. neither

[1, 2]

1

2

b

a[0, 1]

y

01 2 3

[3, 0]

c

[4, 2]

dp q if both

xp < xqyp < yq

a < ba < d

c < d < <

<


Possible topological sorts

1

2

b

a

y

01 2 3 c

d

a < ba < d

c < d


Topological sort understood

<Here the relation is:

{[a, b], [a, d], [c, d]}1

2

b

a

y

01 2 3 c

d

We are looking for a total order relation t such that ⊆ t<

One of the solutions is:a, b, c, d

a < ba < d

c < d


Final statement of topological sort problem

From a given partial order, produce a compatible total order

where:

A partial order p is compatible with a total order t if and only if

p ⊆ t


From constraints to partial orders

Is a relation defined by a set of constraints, such asconstraints =

{[Dishes, Politics], [Üetliberg, Lunch],[Medicine, Lunch], [Lunch, Dishes]}

always a partial order?

Üetliberg



Powers and transitive closure of a relation

r i +1 = r i ; r where ; is composition

Transitive closurer + = r 1 ∪ r 2 ∪ ... always transitive

Ü

M L DP

r 1

r 2

r 3

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Reflexive transitive closure

r i +1 = r i ; r where ; is composition

Transitive closurer + = r 1 ∪ r 2 ∪ ... always transitive

Ü

M L DP

r 1

r 2

r 3

r 0

Reflexive transitive closure:r ∗ = r 0 ∪ r 1 ∪ r 2 ∪ ... always transitive and reflexive

r 0 = id [X ] where X is the underlying set


Acyclic relation

r + ∩ id [X ] = ∅

before+

id [X]

Ü

M L DP

A relation r on a set X is acyclic if and only if:


Acyclic relations and partial orders

Theorems:Any (strict) order relation is acyclic.

A relation is acyclic if and only if its transitiveclosure is a (strict) order.

(Also: if and only if its reflexive transitive closureis a nonstrict partial order)


From constraints to partial orders

The partial order of interest is before +

before ={[Dishes, Politics], [Üetliberg, Lunch],

[Medicine, Lunch], [Lunch, Dishes]}

Üetliberg



Back to software...


The basic algorithm idea

topsort

p

q

r

s

t

u

v

w

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What we have seen

The topological sort problem and its applicationsMathematical background:

Relations as sets of pairsProperties of relationsOrder relations: partial/total, strict/nonstrictTransitive, reflexive-transitive closuresThe relation between acyclic and order relationsThe basic idea of topological sort

Next: how to do it for Efficient operation (O (m+n) for m constraints & n items)Good software engineering: effective API


Reading assignment for next Monday

Touch of Class, chapter on topological sort: 17


End of lecture 19

19 topological sort - ETH Zse.inf.ethz.ch/.../0001/slides/19_topological_sort_6up.pdf3 Intro. to...

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