Week03 Search

8/9/2019 Week03 Search

1/23

This course material is now made available for public usage.

Special acknowledgement to School of Computing, National University of Singapore

for allowing Steven to prepare and distribute these teaching materials.

Com etitivePro rammin

Week03 ProblemSolvingParadigms(FocusonCompleteSearch)

CS3233 CompetitiveProgramming,

StevenHalim,SoC,NUS


2/23

MiniContest#2+Break+Discussion+Admins

Complete

Search Iterative:(Nested)Loops,Permutations,Subsets

RecursiveBacktracking(NQueens),fromeasyto(very)hard

MeetintheMiddle(BidirectionalSearch)

Readathome willnotbetestedinminicontestA B :

Sometipstospeedupyoursolution

Greedyalgorithms

DivideandConquer(D&C)algorithms

EspeciallyBinarySearchtheAnswertechnique


StevenHalim,SoC,NUS


3/23

Iterative:(Nested)Loops,Permutations,Subsets

ecurs ve ac rac ng ueens , romeasy o very ar

StateSpaceSearch

Meet in the Middle Bidirectional Search

COMPLETESEARCH


StevenHalim,SoC,NUS


4/23

IterativeCom leteSearch

Loops(1)

UVa725 Division

Findtwo5digitsnumbers.t. abcde/fghij=N abcdefghijmustbealldifferent,2


5/23


Loops(2)

Morechallengingvariants:

2

3

4

K

nested

loops Somepruningarepossi e,

e.g.usingcontinue,break,orifstatements


StevenHalim,SoC,NUS


6/23


Permutations

UVa 11742 SocialConstraints

Thereare0


7/23


Subsets

UVa12346 WaterGateManagement

Adamhas1 n20watergatestoletoutwaterwhennecessary,

Yourtaskistomanagetheopeningofthewatergatesinordertoget

ridofat

leastthespecifiedtotalflowrateconditionthatthetotal

amage

costis

minimize IterativeCompleteSearchSolution(Subsets):

Foreachsubset,checkifithassufficientflowrate

Ifitis,checkifthetotaldamagecostofthissubsetissmallerthanthe

overallminimumdamagecost sofar

Ifitis,updatetheoverallminimumdamagecostsofar


StevenHalim,SoC,NUS


8/23



StateSpaceSearch


COMPLETESEARCH


StevenHalim,SoC,NUS


9/23

UVa 750 8QueensChessProblem

Put8queensin8x8Chessboard Noqueencanattackotherqueens

Naveways(TimeLimitExceeded)

Choose8outof64cells 64C8=4Billionpossibilities

Insight1:Putonequeenineachcolumn

88 =17Millionpossibilities:O


StevenHalim,SoC,NUS


10/23

Betterway,recursivebacktracking

Insight2:alldifferent

constraintfortherowstoo

WeputonequeenineachcolumnANDeachrow

Findingavalidpermutationoutof8!possiblepermutations

earc spacegoes own rom = o =

Insight3:maindiagonalandsecondarydiagonalcheck

QueenA(i,j)attacksQueenB(k,l)iff

abs( i - k) == abs( j - l )

Scrutinizethesamplecodeofrecursivebacktracking!


StevenHalim,SoC,NUS


11/23

n r w , , a, , ne oun er o o use g o a var a es

bool pl ace( i nt r , i nt c) {

f or i nt r ev = 0 r ev < c r ev++ check r evi ousl l aced ueens

i f ( r w[ pr ev] == r | | ( abs( r w[ pr ev] - r ) == abs( pr ev - c) ) )

r et ur n f al se; / / shar e same r ow or same di agonal - > i nf easi bl e

r et ur n t r ue; }

voi d backt r ack( i nt c) {

i f ( c == 8 && r w[ b] == a) { / / candi dat e sol , ( a, b) has 1 queen

" "pr n , ne oun er , w ;

f or ( i nt j = 1; j < 8; j ++) pr i nt f ( " %d", r w[ j ] + 1) ;

pr i nt f ( " \ n" ) ; }f or i nt r = 0 r < 8 r ++ / / t r al l ossi bl e r ow

i f ( pl ace( r , c) ) { / / i f can pl ace a queen at t hi s col and r ow

r w[ c] = r ; backt r ack( c + 1) ; / / put t hi s queen her e and r ecur se

} }

CS3233CompetitiveProgramming,

StevenHalim,SoC,NUS


12/23

Maybenot

SeeUVa 11195 AnothernQueenProblem Severalcellsareforbidden

Dothishelps?

ncannowbeaslargeasn=14:O??


StevenHalim,SoC,NUS


13/23

Thischeckisslow:

bool pl ace( i nt r , i nt c) {or n pr ev = ; pr ev c; r ev + c ec pr ev ous y p ace queens

i f ( r w[ pr ev] == r | | ( abs( r w[ pr ev] - r ) == abs( pr ev - c) ) )

r et ur n f al se; / / shar e same r ow or same di agonal - > i nf easi bl e

r et ur n t r ue

Wecanspeedupthispartbyusing2*n1boolean arrays

~. . . .

visualization/recursion.html


StevenHalim,SoC,NUS


14/23

Unfortunatelyno

But

fortunately

there

is

a

better

way

of

using

diagonal

checks


StevenHalim,SoC,NUS


15/23



StateSpaceSearch


COMPLETESEARCH


StevenHalim,SoC,NUS


16/23

UVa11212 Editin aBookRujia

Lius

Problem

Givennequaengt paragrap snum ere rom1ton

Arrangethemintheorderof1,2,...,n

,

youcanpressCtrlX(cut)andCtrlV(paste)severaltimes

Youcannotcuttwicebeforepasting,butyoucancutseveralcontiguous

'

Thequestion:Whatistheminimumnumberofstepsrequired?

Exam le1:Inordertomake{2,4,(1),5,3,6}sorted,

youcancut1andpasteitbefore2 {1,2,4,5,(3),6}

then

cut

3

and

paste

it

before

4

{1,

2,

3,4,

5,

6}

done

, , , , ,

youcancut{3,4,5}andpasteitafter{1,2} {1,2,3,4,5}

orcut{1,2}andpasteitbefore{3,4,5} {1,2,3,4,5}


StevenHalim,SoC,NUS


17/23

Answer:k1

Wherekisthenumberofparagraphsinitiallythewrongpositions

r v a u wronga gor m:

Cutaparagraphthatisinthewrongposition

Afterk1such

cut

paste,

we

will

have

asorted

paragraph

Thelastwrongpositionwillbeinthecorrectpositionatthisstage

Butthismaynotbetheshortestway

Examples: , , , , , , steps

{(5),4,3,2,1} {(4),3,2,1,5} {(3),2,1,4,5} {(2),1,3,4,5}

{1,2,3,4,5} 4steps


StevenHalim,SoC,NUS


18/23

{3,2,1}

Answer:2steps,e.g.

, , , , , , ,

{3,2,(1)} {1,(3),2,} {1,2,3}

5,4,3,2,1

Answer:Only

3steps,

e.g.

{5,4,(3,2),1} {3,(2,5),4,1} {3,4,(1,2),5} {1,2,3,4, 5}

Howabout{5,4,9,8,7,3,2,1,6}?

Answer:

4,

but

very

hard

to

compute

manually Howa out 9,8,7,6,5,4,3,2,1

Answer:5,butveryhardtocomputemanually


StevenHalim,SoC,NUS


19/23

Thereareatmostn!permutationsofparagraphs

Withmaximumn=9,thisis9!or362880

Givenapermutationoflengthn(avertex)

n 2 , ..

Thereare

npossible

pasting

points

(index

k[1..(n(ji+1))])

Therefore,foreachvertex,thereareaboutO(n3)branches

TheworstcasebehaviorifwerunasingleBFSonthisState

Spacegraph

is:

O(V+E)

=O(n!

+n!*n3)

=O(n!*n3)

Withn=9,thisis9!*93 =264539520~265M,TLE(ormaybeMLE)


StevenHalim,SoC,NUS


20/23



StateSpaceSearch


COMPLETE

SEARCH


StevenHalim,SoC,NUS


21/23

DepthLimitedSearch(DLS)+IterativeDLS

A*/IterativeDeepeningA*(IDA*)/MemoryBoundedA*

BranchandBound(BnB)

MaybeinWeek12 or...

WewillnottestanyoftheseinminicontestsproblemA/B


StevenHalim,SoC,NUS


22/23

WehaveseensomeCompleteSearchtechniques

There

are

(several)

others Westillneedlotsofpracticethough

WeskippedGreedyandDivide&Conquerthistime

ReadSection3.33.4byyourself

Wewillnottesttheseonminicontests roblemA B

Nextweek,wewillsee(revisit)thefourthparadigm:


StevenHalim,SoC,NUS


23/23

CompetitiveProgramming2.9, Section3.2+8.2

Steven,

Felix


StevenHalim,SoC,NUS

Week03 Search

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