!"#$%" &'($)"'(#$'*+,&-'+(*$./-'0(.-1,&,2 - - PowerPoint PPT Presentation

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Jan 10, 2023 258 likes •1.02k views

!"#$%" &'($)"'(#$'*+,&-'+(*$./-'0(.-1,&,2 34,-*+-&'(5#46.+" 789( :$2&.(;'*+-+6+"($<(8"#4'$.$02=(5>"%"' !"#$%"

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!"#$%"

&'($)"'(#$'*+,&-'+(*$./-'0(.-1,&,2

34,-*+-&'(5#46.+"

789( :$2&.(;'*+-+6+"($<(8"#4'$.$02=(5>"%"'

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!"#$%"

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!"#$"%#&

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!"#$%&'

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!"#$%&'

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!"#$%&'()*+$%&",-).%/+012$,$"%2

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!"#$%"

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!"#$%

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!"#$"

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!"#$%&'"()*+*!,-).,/0),%(

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!"#$%&'()*)+,"(,-..&'(

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  • !"#$%&'#&("'
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  • !"#$%&'#&("'
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  • !"#$%&'#&("'
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  • !"#$%&'#&("'
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  • !"#$%&'#&("'
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  • !"#$%&'#&("'
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!"#$%&'()*&'+$,-./&'(0

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!"#$%&'()*&'+$,-./&'(0

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!"#$"%&&'($

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!"#$$#%%

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!"#$%&'()*#

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!"#$%&'()*+#(+,'#

  • license may do

code and documentation can do

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!"#$%&#'#()"*#$+

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!"##$%&

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!"#$%&'(#%)*&"+&',,(#+)-(%.) /01"20

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< #=

>?9/?09@

Program model as script

declare variables post constraints (creates propagators) define branching

Solve script

basic search strategy Gist: interactive visual search

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!"#$"%&'(#)*+'%,'-."/01

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< #=

56/0>2?,@A9/A09B

Script is class inheriting from class Space

  • members store variables regarded as solution

Script constructor

  • initialize variables
  • post propagators for constraints
  • define branching

Copy constructor and copy function

  • copy a Script object during search

Exploration takes Script object as input

  • returns object representing solution

Main function

  • invokes search engine
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56/0>2,?@/,5AAB,52/7627/9

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56/0=2,>?/,5@@A,52/7627/9

!"#$%&'()*+($,'(-"#./001 !"#$%&'()*+($,'(-2(34$0/001 &2"#+)#35(263$()7($,'(8 $%322)9(#':,4(:,#(; <)6&=%"$)963$()> 64,.($.('< ?#.@34A443; %8)--)B"+".2)C,4).0()%(..(42 6&=%"$<

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  • array of integer variables

stores solution

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56/0=2,>?/,5@@A,52/7627/9

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  • constructor: initialize

variables, post constraints, define branching

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56/0=2,>?/,5@@A,52/7627/9

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  • copy constructor and

copy function

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56/0=2,>?/,5@@A,-?412/762?/

!"#$%&'"%&#"()*&+$,-.-/)012+34-54-64-7,-8 9#1:;' 3)/<6=,4-")/<>=,4-#)/<?=,4-$)/<@=,4 A)/<B=,4-&)/<C=,4-')/<D=,4-()/<E=,F GG-H&31-I&#31';+#13

  • GG-H&31-J';#I2+#K3
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  • GG-H&31-J';#I2+#K3
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variables created in this script (space)

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< !&

56/0=2,>?/,5@@A,-?412/762?/

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  • GG-H&31-J';#I2+#K3
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8 variables

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take values between 0 and 9 (digits)

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56/0>2,?@/,5AAB,-@412/762@/

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refer to variables by their letters

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56/0>2,?@/,5AAB,-@412/762@/

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  • GG-U';#S2-&*"'-12"-/"11"'3
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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< !=

>?1204@,-?412/30421

Defined in namespace !"#$%" Check documentation for available

constraints

Take script reference as first argument

where is the propagator for the constraint to be posted! script is a subclass of Space (computation space)

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*0493/,=>7320?41,34@,*0493/,-?412/30421

Equations of the form

c1x1 cnxn = d

  • integer constants:

ci and d

  • integer variables:

xi

In Gecode specified by arrays

  • integers

(!"#$%&'( ci

  • variables

(!"#)*%$%%*+,-!"#)*%$%&'( xi

Not only equations

  • !./012,-!./032,-!./041,-!./05.,-!./042,-!./052
  • equality, disequality, inequality (less, greater, less or equal, greater or

equal)

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< %#

56/0=2,>?/,5@@A,-?412/762?/

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  • 99-:2"-/+#";'-"<=;1+&#->=31-2&/$

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  • W
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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< %!

*0493/,=>?/9110@41

Other options for posting linear constraints

are available: minimodeling support

linear expressions Boolean expressions matrix classes

See the examples that come with Gecode

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56/0=2,>?/,5@@A,-?412/762?/

  • !"#$%&'()*+($,'(-."#".,'(%/001
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  • K
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56/0=2,>?/,5@@A,-?412/762?/

!"#$%&'"%&#"()*&+$,-.-/)012+34-54-64-7,-8

  • 99-:';#<2-&*"'-12"-/"11"'3

=';#<2)012+34-/4->?@ABCDA!>EFA%>?4->?@ABCGA%>?,H I

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< %&

=/346.04>

Which variable to choose

given order

!"#$%&'$"(")

smallest size

!"#$%&'$*!+)$,!"

smallest minimum

!"#$%&'$,!"$,!"

How to branch: which value to choose

try smallest value

!"#$%&-$,!"

split (lower first)

!"#$%&-$*.-!#$,!"

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< %=

56/0>2,?@/,5AAB,-@>C04D

!!"#$%&'()*'$("+$("*,$%-%. /0%12$(02$%0345$$, &67(08"/0%12$(02$%039"&:";"/<7*04&67(08"&:"= ,>)<17'04?'6-&8"&67(08"&>,:@ A !!"B0(+$(C"*$<3-%."1)(-%."*,$%-%. D-(')7,"/<7*0?"*$<345$$, &67(0:"= (0')(%"%0E"/0%12$(02$%034&67(08?'6-&:@ A

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56/0>2,?@/,5AAB,-@>C04D

!!"#$%&'()*'$("+$("*,$%-%. /0%12$(02$%0345$$, &67(08"/0%12$(02$%039"&:";"/<7*04&67(08"&:"= ,>)<17'04?'6-&8"&67(08"&>,:@ A !!"B0(+$(C"*$<3-%."1)(-%."*,$%-%. D-(')7,"/<7*0?"*$<345$$, &67(0:"= (0')(%"%0E"/0%12$(02$%034&67(08?'6-&:@ A

update all variables needed for solution

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56/0>2,?@/,5AAB,-@>C04D

!!"#$%&'()*'$("+$("*,$%-%. /0%12$(02$%0345$$, &67(08"/0%12$(02$%039"&:";"/<7*04&67(08"&:"= ,>)<17'04?'6-&8"&67(08"&>,:@ A !!"B0(+$(C"*$<3-%."1)(-%."*,$%-%. D-(')7,"/<7*0?"*$<345$$, &67(0:"= (0')(%"%0E"/0%12$(02$%034&67(08?'6-&:@ A

create a new copy

  • f the space

during cloning

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< %=

  • >?@04A

Required during exploration

before starting to guess: make copy when guess is wrong: use copy discussed later

Copy constructor and copy function needed

copy constructor is specific to script updates (copies) variables in particular

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  • =>?,-=412/762=/,@4A,-=>?,B74620=4

Always same structure Important!

must update the variables of a script!

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56/0=2,>?/,5@@A,B/042,C74620?4

  • !!"#$%&'"()*+'%)&

,)%-".$%&'/,)%-0"1 ('-223)+' 44"*"44"('-225&-*6 7

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57==3/>?,56/0@2

Variables

declare as members initialize in constructor update in copy constructor

Posting constraints Create branching Provide copy constructor and copy function

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!"#$%&'(!)*%+,-

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< ))

=>3083?89,593/6.,@4A0491

Returning solutions one by one for script

!"#

depth-first search

$%$

branch-and-bound

&'()*+),-.!#

Interactive, visual search

/0()

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=304,=92.>?@,A0/12,5>8720>4

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

34"567'467"48-+$+9+"4:+34"567'467"48; <=3>34"567'467"48?+4&$1; 54@4#4+$; !A+&34"567'467"48-+B+9+4C"4D#&11+2 BE?F'!"#&1;+54@4#4+B; G '4#H'"+I; G

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  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

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create root space for search

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>304,>92.?@A,B0/12,5?8720?4

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

34"567'467"48-+$+9+"4:+34"567'467"48; <=3>34"567'467"48?+4&$1; 54@4#4+$; !A+&34"567'467"48-+B+9+4C"4D#&11+2 BE?F'!"#&1;+54@4#4+B; G '4#H'"+I; G

create search engine (takes clone of $)

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>304,>92.?@A,B0/12,5?8720?4

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

34"567'467"48-+$+9+"4:+34"567'467"48; <=3>34"567'467"48?+4&$1; 54@4#4+$; !A+&34"567'467"48-+B+9+4C"4D#&11+2 BE?F'!"#&1;+54@4#4+B; G '4#H'"+I; G

root space not any longer needed

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SLIDE 62

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>304,>92.?@A,B0/12,5?8720?4

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

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search first solution and print it

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SLIDE 63

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=304,=92.>?@,A88,5>8720>41

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

34"567'467"48-+$+9+"4:+34"567'467"48; <=3>34"567'467"48?+4&$1; 54@4#4+$; :,!@4+&34"567'467"48-+A+9+4B"4C#&11+2 AD?E'!"#&1;+54@4#4+A; F '4#G'"+H; F

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SLIDE 64

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &#

=96>?9,=012

A graphical tool for exploring the search tree

explore tree step by step tree can be scaled double-clicking node prints information: inspection search for next solution, all solutions

Best to play a little bit by yourself

hide and unhide failed subtrees

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SLIDE 65

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=304,>74620?4@,A012

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slide-66
SLIDE 66

!"#$%&'())*#+,$

  • ./.0.10-2

34--.506.-7%8+("#$"9*%&'+:;$)7%38<7%=<> 21

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SLIDE 67

!"#$%&'()$*'+%&",-./

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!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &&

=9>04?9/@,5AABB

Find distinct digits for letters, such that

and !"#$% maximal

&$#' ( !"&) * !"#$%

slide-69
SLIDE 69

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &=

56/0>2,?@/,5AABB

Similar, please try it yourself at home In the following, referred to by

!"#$%&'(%&#")

slide-70
SLIDE 70

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &=

5>8?04@,5AABBC,D/E9/

Principle

  • for each solution found, constrain remaining search for better

solution

Implemented as additional method

!"#$%&'(!)"*(+),-$#&",.+),-$(/0&+12(34(5

  • 6

Argument 3 refers to so far best solution

  • nly take values from 3
  • never mix variables!

Invoked on object to be constrained

slide-71
SLIDE 71

!""#$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &=

>/?9/,@A/,5BBCC

!"#$%&'(!)"*(+),-$#&",.+),-$(/0&+12(345(6 +),-$(/1,*7)-$7),182(4(9(

  • $&$"+3+&-$:+),-$(/1,*7)-$7),182;.345<

=,$>&# 1.'?@A5B(,.'?CA5B(D.'?EA5B().'?FA5B(8.'?GA5< =,$>&# 431.4H'?@A5B(43,.4H'?CA5B(43D.4H'?EA5B( 43).4H'?FA5B(438.4H'?GA5< ",$ D),18(9(.@IIIIJ43DH!&'.5K@IIIJ43)H!&'.5K@IIJ43,H!&'.5K @IJ431H!&'.5K438H!&'.55< 0)-$.J$L"-B(@IIIIJDK@IIIJ)K@IIJ,K@IJ1K8 ;(D),185< M

value of any next solution value of current best solution b

slide-72
SLIDE 72

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< &=

>304,>92.?@A,B88,5?8720?41

  • !"# $%!"&!"# %'()*+),%'-+%'(./01+2

34"5678#67"49-+$+:+"4;+34"5678#67"49< =>=?34"5678#67"49@+4&$1< 54A4#4+$< ;,!A4+&34"5678#67"49-+8+:+4B"4C#&11+2 8D@E'!"#&1<+54A4#4+8< F '4#G'"+H< F

slide-73
SLIDE 73

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< ="

>304,?74620@4A,B012

!"#$%&'()*+($,'(-+"./0112 "#/ 34"#5"#/ 46+$7)$1468)46+9:;<)= >(#'?,./?,#(@8)3)A)#(B)>(#'?,./?,#(@C D"./EEF4F53<C '(%(/()3C 6(/&6#)GC H

slide-74
SLIDE 74

!"#"$"%$!& '(!!")$*"!+,-./012034,56.7829+,'-:+,;:< =#

57>>3/?@,5A8B04C

Result-only search engines

  • !"#$%&'&

Interactive search engine

  • ()*+

Best solution search uses constrain-method for

posting constraint

Search engine independent of script and constrain-

method