State Space TreesUP

7/29/2019 State Space TreesUP

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Msc. Jorge Espinosa

[email protected]

Horas y correo de contactoLunes y Martes 10-12 m

P19 211 C Ext 492

Medelln 2013

INGENIERA DEL CONOCIMIENTO


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Vision of the Future

Motivation
http://localhost/var/www/apps/conversion/tmp/scratch_4/video/BBC.Visions.of.the.Future.1of3.The.Intelligence.Revolution.2007.DVBC.XviD.MP3.www.mvgroup.org.avi


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Que es el test de Turing!!!


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4

The mind beaten

by the machine?

Is chess playing a proof of intelligent behaviour?


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MARIANAhttp://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ing

resantes/ingreso-buenos-aires/
http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/http://www.uca.edu.ar/uca/index.php/ingreso/index/es/universidad/ingresantes/ingreso-buenos-aires/


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6

Other examples of success:

Chatbot - Alice: http://www.alicebot.org/


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Clever Bothttp://www.cleverbot.com/
http://www.cleverbot.com/http://www.cleverbot.com/


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Siri

http://www.youtube.com/watch?v=0ubSbju9u68&feature=watch_response
http://www.youtube.com/watch?v=0ubSbju9u68&feature=watch_responsehttp://www.youtube.com/watch?v=0ubSbju9u68&feature=watch_response


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Existor

www.existor.com
http://www.existor.com/http://www.existor.com/


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10

Are Mariana/Alice/Clever

Bot/Siri/Existor intelligent?

ABSOLUTELY NOT ! ~ 50000 fairly trivial input-response rules.

+ some pattern matching

+ some knowledge

+ some randomness

NO reasoning component

BUT: demonstrates human-like behaviour.

Won the turing award


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11

Data mining:

An application of Machine Learningtechniques

It solves problems that humans can not solve,because the data involved is too large ..

Detecting cancer

risk molecules is

one example.


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EE.UU. inicia prueba de un ao con

"automviles inteligentes"


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14

Many other applications:

In language and speech processing:

In robotics:

Computervision:


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15

About intelligence ...

When would we consider a program

intelligent ?When do we consider a creative activity of humans to

require intelligence ?

Default answers : Never? / Always?


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What is Turing test!!!


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17

The Turing Test

The long term goal:


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18

The meta-Turing test

The meta-Turing test counts a thing as intelligentifit seeks to devise and apply Turing tests

to objects of its own creation.-- Lew Mammel, Jr.


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19

Artificial Intelligence vs.

Natural Flight


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20

To some extent, we DO simulate:

Artificial Neural Nets:

A VERY ROUGH imitation of a brain structure

Work very well for learning, classifying and pattern

matching.

Very robust and noise-resistant.


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21

Which applications are easy ?

For very specialized, specific tasks: AI

Example:ECG-diagnosis

For tasks requiring common sense: AI


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22

Modeling Knowledge

and managing it .

The LENAT experiment:

15 years of work by 15 to 30 people, trying to model

the common knowledge in the word !!!!

Knowledge should be learned, not engineered.

AI: are we only dreaming ????


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23

But at the end What is Artificial Intelligence?

In Engineering and Computer Science: The development and the study of advanced computer

applications, aimed at solving tasks that - for themoment - are still better preformed by humans.

Well, Artificial Intelligence is ...


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Msc. Jorge Espinosa

[email protected]

Horas y correo de contactoLunes y Martes 10-12 m

P19 211 C Ext 492

Medelln 2013



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25

State-Space representation

and Production Systems

Introduction: what is State-space representation?

What are the important trade-offs?

(E.Rich, Chapt.2)

Basis search methods.

(Winston, Chapt.4 + Russel&Norvig)

Optimal-path search methods.

(Winston, Chapt.5 + Russel&Norvig)

Advanced properties and variants.

(Rich, Chapt.3 + Russel&Norvig + Nilsson)Game Playing

(Winston, Chapt.6 + Rich + Russel&Norvig )


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26


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State-space representation:

Introduction and trade-offs

What is state-space representation?

Which are the technical issues that arise in thatcontext?

What are the alternatives that the paradigm offersto solve a problem in the state-spacerepresentation?


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Example: the 8-puzzle.

Given: a board situation for the 8-puzzle:

1 3 8

2 7

5 4 6

1 2 3

567

48

Problem: find a sequence of moves (allowed under the rules of the

8-puzzle game) that transform this board situation in a desired goal

situation:


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http://mypuzzle.org/sliding

http://www.permadi.com/java/puzzle8/puzzle8a.html
http://mypuzzle.org/slidinghttp://www.permadi.com/java/puzzle8/puzzle8a.htmlhttp://www.permadi.com/java/puzzle8/puzzle8a.htmlhttp://mypuzzle.org/sliding


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State-space representation:

general outline:

Select some way to represent states in the problem in

an unambiguous way.

Formulate all actions that can be performed in states:

including their preconditions and effects== PRODUCTION RULES

Represent the initial state (s).

Formulate precisely when a state satisfies the goal of our

problem. Activate the production rules on the initial state and its

descendants, until a goal state is reached.


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Initial issues to solve:

1 3 82 75 4 6

How to represent states? (repr.1)

Ex.: using a 3 X 3 matrix

How to formulate production rules? (repr. 2) Ex.:

express how/when squares may be moved?

Or: express how/when the blank space is moved?

When is a rule applicable to a state? (matching) How to formulate when the goal criterion is satified and

how to verify that it is?

How/which rules to activate? (control)


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The (implicit) search tree

1 3 8

2 7

5 4 6

1 3 8

2 7

5 4 6

1 3 8

2 7

5 4 6

13

8

2 7

5 4 6

1 3 8

2 7

5

4

6

9!/2

nodes!

goal

Each state-space representation defines a search tree:

8 puzzle= 181.444

15 Puzzle = 650.000.000

24 Puzzle = 5000.000.000.000.000.000.0000.000


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The (implicit) search tree

1 3 8

2 7

5 4 6

1 3 8

2 7

5 4 6

1 3 8

2 7

5 4 6

13

8

2 7

5 4 6

1 3 8

2 7

5

4

6

9!/2

nodes!

goal But this tree is only IMPLICITLY available !!

Each state-space representation defines a

search tree:


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A second example: Chess

Problem: develop a program that plays chess (well).

341

23

45

678

A B C D E F G H

1. A way to represent board situations in an

unambiguous way:

Ex.:

List:

(( king_black, 8, C),

( knight_black, 7, B),

( pawn_black, 7, G),

( pawn_black, 5, F),

( pawn_white, 2, H),( king_white, 1, E))


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35

1

23

4

A B C D E F G H

( (pawn_white,2,x) , (blank, 3, x), (blank, 4, x) )

add( ( pawn_white, 4, x) ),

remove( (pawn_white, 2, x) )

Chess (2):

2. Describe the rules that represent allowedmoves:

Ex.:


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Chess (3):

( (pawn_white,2,x) , (blank, 3, x), (blank, 4, x) )

add( ( pawn_white, 4, x) ),

remove( (pawn_white, 2, x) )

List:

(( king_black, 8, C),

( knight_black, 7, B),

( pawn_black, 7, G),( pawn_black, 5, F),

( pawn_white, 2, H),

( king_white, 1, E))12345

678

A B C D E F G H

Matching mechanism !!

3. Provide a way to check whether a rule isapplicable to some state:

Ex.:


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Chess (4):

37

win( black ) attacked( king_white ) and

no_legal_move( king_white )

+ similar definitions for:

attacked( Piece )

no_legal_move( Piece ) ...

4. How to specify a state in which the goal isreached (= a winning state):

Ex.:


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Chess (5):

?- win( black ).

win( black ) attacked( king_white ) and

no_legal_move( king_white )

Need a theorem prover ( e.g. Prolog) to verify

that the state is a winning one.

5. A way to verify whether a winning state isreached.

Ex.:


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Chess (6).

A program that is ABLE to play chess.

The control problem !

Main focus of this entire

chapter of the course !!

7. A mechanism that selects in each state an appropriate rule to

apply.

6. The initial state.

Chess (7)


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Chess (7).

Move 1

Move 2

Move 3

Implicit search tree

~15

~ (15)2

~ (15)3

Need very efficient search techniques to find good

paths in such combinatorial trees.

Very many issues


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Very many issues

and trade-offs:

41

1. How to choose the rules?2. Should we search through the implicit tree or through an

implicit graph?

3. Do we need an optimal solution, or just any solution?

optimal path problems

4. Can we decompose states into components on which

simple rules can in an independent way?

Problem reduction or decomposability

5. Should we search forwards from the initial state, or

backwards from a goal state?

1 Ch i f th l


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1. Choice of the rules

Example: The water jugs problem:

Given: 2 jugs:

Problem: fill the 4 l jug with 2 l of water.

42

4 l 3 l

Representation:

a state:

[content of large jug, content of small jug]

initial state:

[ 0, 0 ]

goal state:

[ 2, x ]


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Rules for the jugs example:

Fill large:

[ x, y ]and x d> 0 [ x - d, y ]

f


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Rules for the jugs

example (2):

44

Fill large from small:

[ x, y ] and x + y 4 and y > 0[ 4, y-(4-x)]Empty Small in large.

[ x, y ] and x + y 4 and y > 0 [ x + y , 0 ]

Part of the state space:


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45

[ 0, 0 ]

[ 0, 3 ]

Fill small

[ 3, 0 ]

Empty small

in large

[ 3, 3 ]

Fill small

[ 4, 2 ]Fill large

from small


[ 4, 0 ] [ 0, 0 ]

[ 0, 3-d1]

[ 4, 3][ 0, 0]

[ 0, 3-d1-d2]

[ 3-d1, 0]

[ 4, 3-d1]Redundant

subspace !

*

*

[ 2, 0 ]Empty small in large

[ 0, 2 ] Empty large

R l f th j l


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Rules for the jugs example:

Fill large:[ x, y ] and x d> 0 [ x - d, y ] Empty (remove some from) small.

*


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[ 0, 0 ]

[ 0, 3 ]

[ 3, 0 ]

[ 3, 3 ]

[ 4, 2 ]

[ 0, 2 ]

[ 2, 0 ]

Fill small

Empty small

in large

Fill small

Fill large

from small

Empty large

Empty small in large

[ 0, 0 ]

[ 0, 3 ]

[ 0, 0 ]

LOOP !


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2. Avoiding loops:

search in trees or graphs ?

48

[ 0, 0 ]

[ 0, 3 ][ 4, 0 ]

Avoids generating loops: but needs to keep track

of ALL the nodes.

[ 1, 3 ] [ 4, 3 ] [ 3, 0 ]

[ 1, 0 ] [ 3, 3 ]

...

3 Any path versus shortest


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3. Any path, versus shortest

path, versus best path:

Ex.: 8-puzzle: any or shortest path problem.Ex.: Traveling salesperson problem:

Find a sequence of cities ABCDEA such thatthe total distance is MINIMAL.

Boston

Miami

NewYork

SanFrancisco

Dallas

3000

250

1450

1200 1700

33002900

1500

1600

1700

Best path problem

State space


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State space

representation: State:

the list of cities that are already visited

Ex.: ( NewYork, Boston)

Initial state:

Ex.: ( NewYork )

Rules: add 1 city to the list that is not yet a member

add the first city if you already have 5 members

Goal criterion:

first and last city are equal

50

Boston

Miami

NewYork

SanFrancisco

Dallas

0


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51

( NewYork )

( NewYork,

Boston )

( NewYork,

Miami )

( NewYork,

Dallas )

( NewYork,

Frisco )

( NewYork,

Boston,

Miami )

( NewYork,

Frisco,

Miami )

250 1200 15002900

0

250 1200 1500 2900

1450 3300

1700 6200

Keep track ofaccumulated costs in each state if you want

to be sure to get the best path.

4. Problem reduction


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or problem decomposition:

Ex.: Computing symbolic integrals:

State: the integral to compute Rules: integration reduction rules

Goal: all integrals have been eliminated

AND-OR-tree search

x2 + 3x + sin2x . cos2x dxx2 dx 3xdx sin2x . cos2x dx

x3/3 3 xdx((sin2x)/2)2dx (1 - cos2x).cos2x dx

AND

... ... ... ...

Necessary for decomposition:


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Necessary for decomposition:

independence of states:

Ex.: Blocks world problem.

Initially: C is on A and B is on the table.

Rules: to move any free block to another or to the table

Goal: A is on B and B is on C.

53

A

C

B

Goal: A on B and B on C

A

C

B

Goal: A on B

A

C

B

Goal: B on C

AND-OR-tree?

AND


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54

A

C

B

Goal: A on B

A

C

B

Goal: B on C

AND

AC B

Goal: A on BA

C

B

A

C B

But: branches cannot be

combined

5. Forward versus


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backward reasoning:

55

Initial states

Goal states

Forward reasoning (or forward chaining): from initial states to goal states.

5. Forward versus


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backward reasoning:

Initial states

Goal states

Backward reasoning (or backward chaining): from goal states to initial

states.

Criteria:


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Criteria:

57

1 3 8

2 7

5 4 6

1 2 3

567

48

In this case: even the same rules !!

Sometimes equivalent: (Ex.: 8-puzzle:)

Branching factor (Ex.: see previous slide)

Sometimes: no way to start from the goal states

because there are too many (Ex.: chess)

because you cant (easily) formulate the rules in2 directions.

Criteria (2):


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Other possibility: middle-out reasoning.58

The ratio of initial states versus goal states:

Ex.: finding your way to some destination

Backward reasoning!

Providing explanation facilities:

(Ex.: expert systems with user interaction)


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Definition:Programs that implement approaches to search

problems using the state space representation.

Production-rule systems:

Consist of:

A rule base,

a working memory, containing the state(s) that have currently

been reached,

a control strategy for selecting the rules to apply next to the

states in the working memory (including techniques for

matching, verifying preconditions and whether a goal state hasbeen reached)


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Msc. Jorge Espinosa

[email protected]

Horas y correo de contactoMircoles de 9 a 11 a.m.

y Jueves de 8 a 9 a.m.

P19 211 C Ext 492

Medelln 2013


Intro to Graph and trees


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Aim of these slides

Introduce the technical language of graphsand trees.

Motivation:

divide and conquer:

become familiar with the jargon of graphs before

trying to learn about search


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Graphs

Graphs are the mathematical and computerscience abstraction that capture many shared andcommon concepts of real-life objects such as

networks, e.g. water

electricity

internet

mazes road maps

timetabling

IA Games Theory


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Graphs

Many problems in CS/AI can be reduced toproblems in graph theory, gives a good way to share results, methods &

implementations across many areas Graph Theory has its own jargon; but it is worth

becoming familiar with it e.g. in discussions of network topology, social

networks,


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Terminology of Graphs

Just giving the ideas. Exact mathematical definitions notneeded for this course, and are available in many textbooks,wikipedia, etc.

A Graph consists of a set V of nodes, and a set E ofedges

Node has a unique label for identification

Edge connects two nodes any two nodes have at most one edge between them

usually forbid self-edges : edges from a node back to itself can be a

directed edge: e.g. from A to B a one-way street undirected edge: e.g. between A and B a standard street


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Graph Example

This is not a map!

The positions of thenodes do not matter,

only theirinterconnections

E.g. LondonUnderground Mapgives

interconnections, butnot true locations

B

C

D

E

F

G

A


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Mapas


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Mapas


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Mapa


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Graph Example

Crossing of edges(arcos,aristas) hasno meaning

A-D and B-E crossin the picture butthey do notinterconnect

These two

pictures are thesame graph

B

CD

E

F G

A

B

CD

E

F

G

A


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Terminology of Graphs

Path

connected sequence of edges (arcos-aristas):

e.g. A to B to F to C to G

usually not allowed to use the same node (or edge)twice

if the edges are directed then the path has to followthe directions of the edges. E.g. follow the one-waystreets on a map

B

CD

E

F

G

A


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Reachable

Node B is said to be reachable from node Aif and only if

there is a path from A to B

Relevance to real-world: many search problems are questions about whether or

not some goal is reachablefrom some start node part of maintaining the internet topology is ensuring

that any node (site) is reachable from any other site a design factor in the internet was that nodes stayreachable even if some links are broken

e.g. in the event of nuclear war


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Connected Graphs

Connected Graph. Definition: for any two nodes there is a path between them

i.e. any node is reachable from any other node

E.g. this graph is connected

B

CD

E

F

G

A


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Connected Graphs

E.g. this graph is disconnected

There is no path from A to B

B

CD

E

F

G

A


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Cycles

Definition: a cycle is a path that goes in a loopfrom a node back to itself, and without usingthe same node twice

E.g. A-B-E-A in

B

CD

E

F

G

A


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Tree

A tree is a graph that is connected

but becomes disconnected if you remove any

edge (cut the edge) Matches nature

real trees are connected

cut any branch then the tree falls into two pieces


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Tree: Example

This is a tree

B

CD

E

F

G

A

H

I

J


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Acyclic Graphs

Defn: An acyclic graph has no cycles

Alternative definition:

A tree is a graph that is

connected, and

acyclic


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Trees

Question: Why is a acyclic equivalent tobecomes disconnected on any cut?

B

CD

E

F

G

A

Suppose we cut edge A-B

Graph stays connected


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Trees

A Tree is connected: given any two nodes A and B there exists at least one pathbetween them Question: Given a tree, and any two nodes A and B, is the path between them

unique? or might there be pairs of nodes that have more than one path betweenthem?

Suppose nodes E and F have two distinct paths P1 and P2 between them: Using P1 followed by reverse(P2), we would be able to construct a cycle. But trees are acyclic, hence, paths between nodes of a tree must be unique

B

D

E

FA

P1(E,F)

B

D

E

FA

P2(E,F)

B

D

E

FA

P1(E,F) U P2(F,E)


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Trees

A tree is a graph that:1. is connected but becomes disconnected on removing

any edge

2. is connected and acyclic

3. has precisely one path between any two nodes

The above 3 definitions are equivalent

Suggestion: become comfortable enough with the

definitions for this equivalence to be self-evident

R d T


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Rooted Trees

Often in trees have a special node called theroot

E.g.

J

B

CD

E

F

G

A

H

I

R d T


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Rooted Trees

Often in trees have a special node called the root Usually also give edges a direction away from the root

e.g.

J

B

CD

E

F

G

A

H

I

R d T


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Rooted Trees

Often, on making a picture of a tree Pick up the tree by the root

let it hang, and shake it a bit

JB C

D

E

F

G

A

H

I

J

B

CD

E

F

G

A

H

I

But in practice

you dont usually get the pretty version

programs have to use messy representations

R t d T J


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Rooted Trees: Jargon

The nodes directly below a node arecalled its children e.g. H and E are children of B B is the parent of H H and E are siblings

Children, children of children, etc aredescendents D is a descendent of B

Parents,parents of parents, etc areancestors B is an ancestor of D

A node without children is called aleaf node (or terminal node) e.g. G is a leaf node

JB C

D

E

F

G

A

H

I

R t d T E l


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Rooted Tree : Example

File systems are rooted trees (Suggestion: When thinking/studying search and it becomes

too abstract, then filesystems provide a concrete example thatmight make it easier to understand)

DOS/Windows C:\ is the root (of the C drive)

Unix / is the root there is even a root user the one that has enough privileges to

modify the root directory (and all others)

R t d T E l


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File systems are rootedtrees

nodes = files or folders

(a.k.a. directories)

edges = links from a folderto the files/folders that itcontains

links are addresses on the

hard drive

\J

\B

\B\E

\

\B\E\A

\B\H

\B\E\I

R t d T E l


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Actual layout on hard drive will be a mess

\J \B \B\E\ \B\E\A\B\H \B\E\I

R t d T E l


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Note: there might be a lot of data on the hard drivethat has no valid link to it

removing a file means just remove the link to it

can still be on the hard-drive but you cannot reach it byfollowing links from the root directory

scanning the hard drive for a string does not tell youwhether the string occurs in the filesystem

E.g. del \B\E\A (or unix rm /B/E/A) gives

\J \B \B\E\ \B\E\A\B\H \B\E\I

S ft f R


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Software for Recovery

T B hi f t


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Trees: Branching factor

The branching factor of a node is just the number ofbranches emerging from it its number of children

Branching factor of the tree itself is (usually): largest

branching factor of any node

(Might also read about an effective branchingfactor some form of average over the nodes)

Trees b 2


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Trees: b=2

If the largest branchingfactor is two then wehave a binary tree

JB

E

F

A

H

I

Trees b 1


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Trees: b=1

If the largest branchingfactor is one then we havea linked list

These are often used inprogramming as acontainer: a way to

store a collection of

objects

B

E

F

A

Aside: Doubly Linked Lists


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Aside: Doubly Linked Lists

Take a linked list and alsoadd edges from each(non-root) node to itsparent

Used in programming as acontainer:

can easily walk in both

directions

B

E

F

A

Trees: Depth


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Trees: Depth

The depth, d, of a node isjust the number of edgesit is away from the rootnode

The depth of a tree is thedepth of the deepestnode

in this case, depth=4

JB C

D

E

F

G

A

H

I

d=0

d=1

d=2

d=3

d=4

Tree Sizes


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Tree Sizes

Suppose we have a tree with branching factor b

depth d

What is the maximum number of nodes doesit can have?

Sizes of Trees: Branching Factor


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Sizes of Trees: Branching Factor

Increasing b rapidly increases the tree size:

d nodes at d, b=2, 2d nodes at d, b=3, 3d

0 1 11 2 3

2 4 9

3 8 27

4 16 81

5 32 243

6 64 729

Sizes of Trees


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Sizes of Trees

Pattern seen in binary tree:nodes at d or less = nodes at d+1 1

= 2d+1-1 = O(2d)

The number of nodes grows exponentially Another manifestation of the Combinatorial Explosion

Similarly, for a general tree number of nodes is O(bd)

but, remember big O gives an upper bound,

real trees might have a lot fewer nodes

Trees are generally very leaf-heavy a large fraction of thenodes are leaves e.g. on your file system you probably have far more files

than folders

Effects of Tree Sizes


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Effects of Tree Sizes

If possible try to work with trees with smaller branching factor

smaller depth

Be careful when programming that yourmemory requirements do not explode

Summary


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Summary

Definitions of graphs

path

connected cycles, and acyclic

tree, and rooted tree

Trees grow exponentially with depth

Exercise


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Exercise

Given a tree with Branching factor of 4, howmany nodes are acumulated at level 5(d=5)?

If the b= 5, how many nodes are acumulatedat level 6 (d=6)??

Questions?


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Questions?


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