Lecture 19: Topological sort Part 1: Problem and math basis by - - PowerPoint PPT Presentation

Chair of Softw are Engineering Chair of Softw are Engineering

Einführung in die Programmierung Einführung in die Programmierung Introduction to Programming

P f D B t d M

• Prof. Dr. Bertrand Meyer

Lecture 19: Topological sort Part 1: Problem and math basis

by Caran d’Ache

Introduction to Programming, lecture 19: Topological sort

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“Topological sort”

From a given partial order, produce a compatible total order compatible total order

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The problem

From a given partial order, produce a compatible total order

Partial order: ordering constraints between elements of a set, e.g.

produce a compatible total order

Partial order ordering constraints between elements of a set, e.g.

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

g

“Take your medicine before lunch” “Finish lunch before removing dishes”

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

Üetliberg, Medicine, Lunch, Dishes, Politics : OK Medicine, Üetliberg, Lunch, Dishes, Politics : OK

P l M d L h D h Ü l b K

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

Why we are doing this!

Very common problem in many different areas Interesting efficient non-trivial algorithm Interesting, efficient, non-trivial algorithm Illustration of many algorithmic techniques Illustration of data structures complexity (big Oh Illustration of data structures, complexity (big-Oh

notation), and other topics of last lecture

Illustration of software engineering techniques: Illustration of software engineering techniques:

from algorithm to component with useful API

Opportunity to learn or rediscover important

Opportun ty to learn or red scover mportant mathematical concepts: binary relations (order relations in particular) and their properties

It’s just beautiful!

Today: problem and math basis

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Next time: detailed algorithm and component

Reading assignment for next Monday

Touch of Class, chapter on topological sort: 17

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Topological sort: example uses

From a dictionary, produce a list of definitions such that no word From a d ct onary, produce a l st of def n t ons such that no word

• ccurs prior to its definition

P d l h d l f i f k Produce a complete schedule for carrying out a set of tasks, subject to ordering constraints (This is done for scheduling maintenance tasks for industrial tasks,

• ften with thousands of constraints)

Produce a version of a class with the features reordered so that no feature call appears before the feature’s declaration

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Rectangles with overlap constraints

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

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Rectangles with overlap constraints

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

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The problem

From a given partial order, produce a compatible total order

Partial order: ordering constraints between elements of a set, e.g.

produce a compatible total order

Partial order ordering constraints between elements of a set, e.g.

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

g

“Take your medicine before lunch” “Finish lunch before removing dishes”

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

Üetliberg, Medicine, Lunch, Dishes, Politics : OK Medicine, Üetliberg, Lunch, Dishes, Politics : OK

P l M d L h D h Ü l b K

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

Pictured as a graph

Üetliberg Lunch Dishes Politics Medicine

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

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Sometimes there is no solution

“Introducing recursion requires that

Introduc ng recurs on requ res that students know about stacks”

“You must discuss abstract data

types before introducing stacks types before introducing stacks

“Abstract data types rely on

recursion”

The constraints introduce a cycle

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The constraints introduce a cycle

Overall structure (1)

Given: class ORDERABLE [G] feature A type G A set of elements class ORDERABLE [G] feature elements: LIST [G ]

• f type G

constraints: LIST [TUPLE [G, G ]] A set of constraints between these elements [ ] R i d topsort : LIST [G ] is between these elements Required: An enumeration of the elements, in an order l h h ... ensure compatible (Result, constraints) compatible with the constraints end

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Some mathematical background...

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Binary relation on a set

Any property that either holds or doesn’t hold between two elements of a set two elements of a set On a set PERSON of persons example relations are: On a set PERSON of persons, example relations are:

mother : a mother b holds if and only if a is the

mother of b

father : child : sister : sibling : (brother or sister)

Note: relation names will i

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

appear in green

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Example: the before relation

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

The set of interest:

Tasks = {Politics, Lunch, Medicine, Dishes, Üetliberg} Tasks {Politics, Lunch, Medicine, Dishes, Üetliberg}

The constraining relation:

Dishes before Politics Üetliberg before Lunch g Medicine before Lunch Lunch before Dishes

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

See examples on next slide

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Example: the before relation

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

The set of interest:

F n sh lunch before remov ng d shes

elements = {Politics, Lunch, Medicine, Dishes, Üetliberg} The constraining relation: before = {[Dishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, Dishes]}

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Using ordinary set operators

spouse = wife ∪ husband sibling = sister ∪ brother ∪ id [Person] sibling = sister ∪ brother ∪ id [Person] sister ⊆ sibling father ⊆ ancestor universal [X ] = X x X (cartesian product) [X] empty [X] = ∅

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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)) R fl i Reflexive: a r a Irreflexive: not (a r a) S t i b b Symmetric: a r b ⇒ b r a Antisymmetric: (a r b) ∧ (b r a) ⇒ a = b A t i t (( b) (b )) Asymmetric: 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

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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

Reflexive, symmetric, transitive

sister

Symmetric, irreflexive Reflexive, antisymmetric

family_head

(a family_head b means a is the head

• f b’s family, with one head per family)

mother

Asymmetric irreflexive

Total: (a ≠ b) ⇒ (a r b) ∨ (b r a) Reflexive: a r a mother

Asymmetric, irreflexive

Reflexive: a r a Irreflexive: not (a r a) Symmetric: a r b ⇒ b r a Symmetric: a r b ⇒ b r a Antisymmetric: (a r b) ∧ (b r a) ⇒ a = b Asymmetric: not ((a r b) ∧ (b r a))

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y (( ) ( )) Transitive: (a r b) ∧ (b r c) ⇒ a r c

Total order relation (strict)

Relation is strict total order if: T l

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

• Total
• Irreflexive
• Transitive

Irreflexive: not (a r a) Symmetric: a r b ⇒ b r a Asymmetric: not ((a r b) ∧ (b r a))

• Transitive

Transitive: (a r b) ∧ (b r c) ⇒ a r c

Example: “less than” < on natural numbers

0 < 1 0 < 2 0 < 3 0 < 4

1 2 3 4 5

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

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,

...

Theorem

A strict order is asymmetric (total)

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Total order relation (strict)

Relation is strict total order if: T l

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a)) Irreflexive: not (a r a) Total Irreflexive Transitive Irreflexive: not (a r a) Symmetric: a r b ⇒ b r a Asymmetric: not ((a r b) ∧ (b r a)) Transitive Transitive: (a r b) ∧ (b r c) ⇒ a r c

Theorem: A strict order is asymmetric total

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Total order relation (non-strict)

Relation is non-strict total order if: Total

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a)) I fl i t ( )

Total Reflexive Transitive A i i

Irreflexive: not (a r a) Symmetric: a r b ⇒ b r a Antisymmetric: (a r b) ∧ (b r a) ⇒ a = b Transitive:(a r b) ∧ (b r c) ⇒ a r c

Antisymmetric

Transitive:(a r b) ∧ (b r c) ⇒ a r c

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

1 2 3 4 5

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

1 2 3 4 5

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

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2 ≤ 2 2 ≤ 3, 2 ≤ 4, ...

...

Total order relation (strict)

Relation is strict total order if: T l

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

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Partial order relation (strict)

Relation is strict partial order if: T l

Total: (a ≠ b) ⇒ ((a r b) ∨ (b r a)) Total Irreflexive Transitive Irreflexive: not (a r a) Symmetric: a r b ⇒ b r a Antisymmetric: (a r b) ∧ (b r a) ⇒ a = b Transitive Antisymmetric: (a r b) ∧ (b r a) ⇒ a = b Transitive:(a r b) ∧ (b r c) ⇒ a r c [1, 2]

2

b

y

[4, 2]

d Example: relation between

1

b

[0, 1]

d p points in a plane: p q if both < a

x

[3, 0]

xp < xq yp < yq

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x

1 2 3 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

[1, 2]

2

y

[4, 2]

p q if both

<

1

b

[0, 1]

d

p q if both xp < xq y < y

<

1

a

[3, 0]

yp < yq

Here the following hold: No link between a and c b and c:

1 2 3 c

Here the following hold: c a No link between a and c, b and c: a c nor e.g. neither

a < b c < d < < a < d

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Possible topological sorts

2

y

1

b d

1

a

1 2 3 c

a < b c < d a < d

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Topological sort understood

2

y

< Here the relation is: {[a, b], [a, d], [c, d]}

1

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

1

a One of the solutions is: b d

1 2 3 c

a, b, c, d We are looking for a total order relation t such that

⊆ t

< a < b c < d relation t such that

⊆ t

< a < d

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Final statement of topological sort problem From a given partial order, produce a compatible total order compatible total order where: A partial order p is compatible with a A partial order p is compatible with a total order t if and only if p ⊆ t

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From constraints to partial orders

Is a relation defined by a set of constraints, such as y constraints = {[Dishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, Dishes]} always a partial order?

Üetliberg Medicine Lunch Dishes Politics

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Powers and transitive closure of a relation

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

Ü P M L D P

Transitive closure r + = r 1 ∪ r 2 ∪ always transitive

r 1

r r ∪ r ∪ ... always transitive

r r 2 r 3

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Reflexive transitive closure

r i +1 = r i ; r where ; is composition r 0 = id [X ] where X is the underlying set p

Ü P M L D P

Transitive closure r + = r 1 ∪ r 2 ∪ always transitive

r 1 r 0

r r ∪ r ∪ ... always transitive

r r 2 r 3

Reflexive transitive closure:

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

Acyclic relation

r + ∩ id [X ] = ∅ A relation r on a set X is acyclic if and only if:

Ü P M L D P

before + before + id [X]

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Acyclic relations and partial orders

Theorems:

Any (strict) order relation is acyclic.

A l i i li if d l if i i i

A relation is acyclic if and only if its transitive

closure is a (strict) order.

(Also: if and only if its reflexive transitive closure is a nonstrict partial order) p )

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From constraints to partial orders

The partial order of interest is before + before = before = {[Dishes, Politics], [Üetliberg, Lunch], [Medicine, Lunch], [Lunch, Dishes]} Üetliberg Medicine Lunch Dishes Politics

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Back to software...

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The basic algorithm idea

p r t v

• q

s u w

• u
• topsort

topsort

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What we have seen

The topological sort problem and its applications Mathematical background: Mathematical background:

Relations as sets of pairs Properties of relations Properties of relations Order relations: partial/total, strict/nonstrict Transitive, reflexive-transitive closures

,

The relation between acyclic and order relations The 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

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Good software engineering: effective API

Reading assignment for next Monday

Touch of Class, chapter on topological sort: 17

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End of lecture 19

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