Sp14 Cs188 Lecture 5 -- Csps II
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Transcript of Sp14 Cs188 Lecture 5 -- Csps II
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$oday
+,cient Solution o CSPs
-ocal Searc%
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.eminder: CSPs
CSPs: /ariables Domains Constraints
Im(licit 0(rovide code tocom(ute
+2(licit 0(rovide a list o t%e legaltu(les
Unary ) !inary ) 34ary
5oals: 6ere: fnd any solution
Also: fnd all fnd best etc'
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!ac"trac"ing Searc%
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Im(roving !ac"trac"ing
5eneral4(ur(ose ideas give %uge gains in s(eed 7 but its all still 3P4%ard
9iltering: Can &e detect inevitable ailure early
;rdering:
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Arc Consistency and !eyond
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Arc Consistency o an +ntire CSP
A sim(le orm o (ro(agation ma"es sure all arcs are simuconsistent:
Arc consistency detects ailure earlier t%an or&ard c%ec"i Im(ortant: I > loses a value neig%bors o > need to be rec =ust rerun ater eac% assignment?
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-imitations o Arc Consistency
Ater enorcing arcconsistency: Can %ave one solution
let
Can %ave multi(le
solutions let Can %ave no solutions let
0and not "no& it
Arc consistency still
What wenwrong her
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K4Consistency
Increasing degrees o consistency
14Consistency 03ode Consistency: +ac% single nodesdomain %as a value &%ic% meets t%at nodes unaryconstraints
4Consistency 0Arc Consistency: 9or eac% (air o nodesany consistent assignment to one can be e2tended tot%e ot%er
K4Consistency: 9or eac% " nodes any consistentassignment to "41 can be e2tended to t%e "t% node'
6ig%er " more e2(ensive to com(ute
Bou need to "no& t%e " case: arc consistenc
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Strong K4Consistency
Strong "4consistency: also "41 "4 7 1 consistent
Claim: strong n4consistency means &e can solve &it%outbac"trac"ing?
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Structure
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$ree4Structured CSPs
$%eorem: i t%e constraint gra(% %as no loo(s t%e CSP canin ;0n d time Com(are to general CSPs &%ere &orst4case time is ;0dn
$%is (ro(erty also a((lies to (robabilistic reasoning 0latere2am(le o t%e relation bet&een syntactic restrictions and
com(le2ity o reasoning
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$ree4Structured CSPs
Algorit%m or tree4structured CSPs: ;rder: C%oose a root variable order variables so t%at (
(recede c%ildren
.emove bac"&ard: 9or i n : a((ly.emoveInconsistent0Parent0>i>i
Assign or&ard: 9or i 1 : n assign >i consistently &it%
.untime: ;0n d
0&%y
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$ree4Structured CSPs
Claim 1: Ater bac"&ard (ass all root4to4lea arcs are cons Proo: +ac% >→ B &as made consistent at one (oint and B
could not %ave been reduced t%ereater 0because Bs c%ild(rocessed beore B
Claim : I root4to4lea arcs are consistent or&ard assignmbac"trac"
Proo: Induction on (osition
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Im(roving Structure
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3early $ree4Structured CSPs
Conditioning: instantiate a variable (rune its neigdomains
Cutset conditioning: instantiate 0in all &ays a setvariables suc% t%at t%e remaining constraint gra(
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Cutset Conditioning
SA
SA SA
Instantiate t%ecutset 0all (ossible
&ays
Com(ute residualCSP or eac%assignment
Solve t%e residualCSPs 0treestructured
C%oose a cutset
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Cutset @uiH
9ind t%e smallest cutset or t%e gra(% belo&
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$ree Decom(osition
Idea: create a tree4structured gra(% o mega4variables
+ac% mega4variable encodes (art o t%e original CSP
Sub(roblems overla( to ensure consistent solutions
M1 M2 M3 M4
{(WA=r,SA=g,NT=b),
(WA=b,SA=r,NT=g),
…}
{(NT=r,SA=g,Q=b),
(NT=b,SA=g,Q=r),
…}
Agree: (M1,M2) ∈
{((WA=g,SA=g,NT=g), (NT=g,SA=g,Q=g)), …}
Agree
on
sharedvars
NT
SA
WA
Q
SA
NT
Agree
on
sharedvars
NS
W
SA
Q
Agree
on
sharedvars
SA
NS
W
i
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Iterative Im(rovement
I i Al i % CSP
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Iterative Algorit%ms or CSPs
-ocal searc% met%ods ty(ically &or" &it% Jcom(lete statevariables assigned
$o a((ly to CSPs: $a"e an assignment &it% unsatisfed constraints ;(erators reassign variable values 3o ringe? -ive on t%e edge'
Algorit%m:
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+2am(le: F4@ueens
States: F Mueens in F columns 0FF NO states
;(erators: move Mueen in column 5oal test: no attac"s +valuation: c0n number o attac"s
#Demo: n4Mueens itim rovement 0-ND1
/ideo o Demo Iterative Im(rovem
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/ideo o Demo Iterative Im(rovem@ueens
/ideo o Demo Iterative Im(rove
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/ideo o Demo Iterative Im(roveColoring
P =i C Li t
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Perormance o =in4ConLicts
5iven random initial state can solve n4Mueens in almost ctime or arbitrary n &it% %ig% (robability 0e'g' n 1EEEE
$%e same a((ears to be true or any randomly4generatedexcept in a narro& range o t%e ratio
S CSP
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Summary: CSPs
CSPs are a s(ecial "ind o searc% (roblem: States are (artial assignments 5oal test defned by constraints
!asic solution: bac"trac"ing searc%
S(eed4u(s: ;rdering 9iltering Structure
Iterative min4conLicts is oten eQective in
(ractice
-ocal Searc%
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-ocal Searc%
-ocal Searc%
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-ocal Searc%
$ree searc% "ee(s une2(lored alternatives on t%e ringe 0ecom(leteness
-ocal searc%: im(rove a single o(tion until you cant ma"eringe?
3e& successor unction: local c%anges
5enerally muc% aster and more memory e,cient 0but inc
6ill Climbing
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6ill Climbing
Sim(le general idea:
Start &%erever .e(eat: move to t%e best neig%boring state I no neig%bors better t%an current Muit
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6ill Climbing Diagram
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Simulated Annealing
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Simulated Annealing
Idea: +sca(e local ma2ima by allo&ing do&n%illmoves
!ut ma"e t%em rarer as time goes on
3!
Simulated Annealing
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Simulated Annealing
$%eoretical guarantee: Stationary distribution:
I $ decreased slo&ly enoug%
&ill converge to o(timal state?
Is t%is an interesting guarantee
Sounds li"e magic but reality is reality: $%e more do&n%ill ste(s you need to esca(e a
local o(timum t%e less li"ely you are to everma"e t%em all in a ro&
Peo(le t%in" %ard about ridge operators &%ic%let you um( around t%e s(ace in better &ays
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+2am(le: 34@ueens
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+2am(le: 34@ueens
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3e2t $ime: Adversarial Searc