sp14 cs188 lecture 5 -- csps ii

8/15/2019 Sp14 Cs188 Lecture 5 -- Csps II

1/38


2/38

$oday

+,cient Solution o CSPs

-ocal Searc%


3/38

.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'


4/38

!ac"trac"ing Searc%


5/38

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:


6/38

Arc Consistency and !eyond


7/38

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?


8/38

-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


9/38


10/38

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


11/38

Strong K4Consistency

Strong "4consistency: also "41 "4 7 1 consistent

Claim: strong n4consistency means &e can solve &it%outbac"trac"ing?


12/38

Structure


13/38


14/38

$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


15/38


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


16/38


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


17/38

Im(roving Structure


18/38

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(


19/38

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


20/38

Cutset @uiH

9ind t%e smallest cutset or t%e gra(% belo&


21/38

$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


22/38

Iterative Im(rovement

I i Al i % CSP


23/38

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:


24/38

+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


25/38

/ideo o Demo Iterative Im(rovem@ueens

/ideo o Demo Iterative Im(rove


26/38

/ideo o Demo Iterative Im(roveColoring

P =i C Li t


27/38

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


28/38

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%


29/38

-ocal Searc%

-ocal Searc%


30/38

-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


31/38

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


32/38

6ill Climbing Diagram


33/38

Simulated Annealing


34/38

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


35/38

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


36/38

+2am(le: 34@ueens


37/38

+2am(le: 34@ueens


38/38

3e2t $ime: Adversarial Searc

sp14 cs188 lecture 5 -- csps ii

Documents