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PancakesWithAProblem!GreatTheoreticalIdeasInComputerScienceAnupamGuptaCS15-251Fall2O05Lecture1Aug29th,2OO5CarnegieMellonUniversityPancakesWithAProblem!GreatCourseStaffProfs: AnupamGupta JohnLaffertyTAs: KanatTangwongsan YinmengZhang /~15251Pleasecheckthewebpagesregularly!CourseStaffPleasefeelfreetoaskquestions!
((()))Pleasefeelfree((()))CourseDocument
Youmustreadthiscarefully.Gradingformulaforthecourse.40%homework
5%in-classquizzes25%in-recitationtests30%finalSevenpointsadaylatepenalty.Collaboration/CheatingPolicyYoumayNOTsharewrittenwork.Wereusehomeworkproblems.CourseDocument
YoumustreadPancakesWithAProblem!GreatTheoreticalIdeasInComputerScienceAnupamGuptaCS15-251Fall2O05Lecture1Aug29th,2OO5CarnegieMellonUniversityPancakesWithAProblem!GreatThechefatourplaceissloppy,andwhenhepreparesastackofpancakestheycomeoutalldifferentsizes.Therefore,whenIdeliverthemtoacustomer,onthewaytothetableIrearrangethem(sothatthesmallestwindsupontop,andsoon,downtothelargestatthebottom)
Idothisbygrabbingseveralfromthetopandflippingthemover,repeatingthis(varyingthenumberIflip)asmanytimesasnecessary.ThechefatourplaceissloppDevelopingANotation:
TurningpancakesintonumbersDevelopingANotation:
TurningDevelopingANotation:
TurningpancakesintonumbersDevelopingANotation:
TurningDevelopingANotation:
Turningpancakesintonumbers23451DevelopingANotation:
TurningDevelopingANotation:
Turningpancakesintonumbers52341DevelopingANotation:
TurningHowdowesortthisstack?
Howmanyflipsdoweneed?5
2
3
4
1Howdowesortthisstack?
How4FlipsAreSufficient12345523414321523415143254FlipsAreSufficient12345523AlgebraicRepresentation52341X=Thesmallestnumber
offlipsrequiredtosort:?X?UpperBoundLowerBoundAlgebraicRepresentation52341XAlgebraicRepresentation52341X=Thesmallestnumber
offlipsrequiredtosort:?X4UpperBoundLowerBoundAlgebraicRepresentation52341X4Flipsarenecessaryinthiscase523414132514325Flip1hastoput5onbottomFlip2mustbring4totop.4FlipsarenecessaryinthisUpperBoundLowerBound4X4X=4UpperBoundLowerBound45thPancakeNumberThenumberofflipsrequiredtosorttheworstcasestackof5pancakes.?P5
?UpperBoundLowerBoundP5=5thPancakeNumberThenumbero5thPancakeNumberThenumberofflipsrequiredtosorttheworstcasestackof5pancakes.4P5
?UpperBoundLowerBoundP5=5thPancakeNumberThenumbero..........The5thPancakeNumber:
TheMAXoftheX’s120119932X1X2X3X119X120523414..........The5thPanThe5thPancakeNumber:
TheMAXoftheX’s1201993..........X1X2X3X119X1205234141234554321The5thPancakeNumber:
TheM1201993..........X1X2X3X119X1205234141234554321P5=
MAXovers2stacksof5
ofMIN#offlipstosorts
1201993..........X1X2Pn=
MAXovers2stacksofnpancakes
ofMIN#offlipstosorts
Pn=
Thenumberofflipsrequiredtosorttheworst-casestackofnpancakes.Pn=
MAXovers2stacksofnPn=
MAXovers2stacksofnpancakes
ofMIN#offlipstosorts
Pn=
Thenumberofflipsrequiredtosortaworst-casestackofnpancakes.Pn=
MAXovers2stacksofnBeCool.LearnMath-speak.Pn=
Thenumberofflipsrequiredtosorta
worst-casestackofnpancakes.Pn=
ThenumberofflipsrequWhatisPnforsmalln?Canyoudon=0,1,2,3?WhatisPnforsmalln?CanyouInitialValuesOfPnn0123Pn0013InitialValuesOfPnn0123Pn001P3=31
3
2requires3Flips,henceP3≥3.ANYstackof3canbedonebygettingthebigonetothebottom(≤2flips),andthenusing≤1extrafliptohandlethetoptwo.Hence,P3=3.P3=31
3
2requires3nthPancakeNumberThenumberofflipsrequiredtosortaworstcasestackofnpancakes.?Pn
?UpperBoundLowerBoundPn=nthPancakeNumberThenumberoBracketing:
WhatarethebestlowerandupperboundsthatIcanprove?≤f(x)≤[]Bracketing:
Whatarethebest?Pn
?TakeafewminutestotryandproveupperandlowerboundsonPn,
forn>3.?Pn?TakeafewminutestBringbiggesttotop.
Placeitonbottom.Bringnextlargesttotop.
Placesecondfrombottom.Andsoon…Bring-to-topMethodBringbiggesttotop.
PlaceiUpperBoundOnPn:
BringToTopMethodFornPancakesIfn=1,noworkrequired-wearedone!Otherwise,flippancakentotopand
thenflipittopositionn.Nowuse:BringToTopMethod
Forn-1PancakesTotalCost:atmost2(n-1)=2n–2flips.UpperBoundOnPn:
BringToToBetterUpperBoundOnPn:
BringToTopMethodFornPancakesIfn=2,atmostoneflipandwearedone.Otherwise,flippancakentotopand
thenflipittopositionn.Nowuse:BringToTopMethod
Forn-1PancakesTotalCost:atmost2(n-2)+1=2n–3flips.BetterUpperBoundOnPn:
BrinBringtotopnotalwaysoptimalforaparticularstackBring-to–toptakes5flips,butwecandoin4flips3214552341231454132514325Bringtotopnotalwaysoptima?Pn
2n-3WhatotherboundscanyouproveonPn??Pn2n-3Whatotherbounds9
16SupposeastackScontainsapairofadjacentpancakesthatwillnotbeadjacentinthesortedstack.AnysequenceofflipsthatsortsstackSmustinvolveoneflipthatinsertsthespatulabetweenthatpairandbreaksthemapart.BreakingApartArgument9BreakingApartArgument9
16SupposeastackScontainsapairofadjacentpancakesthatwillnotbeadjacentinthesortedstack.AnysequenceofflipsthatsortsstackSmustinvolveoneflipthatinsertsthespatulabetweenthatpairandbreaksthemapart.Furthermore,thissameprincipleistrueofthe“pair”formedbythebottompancakeofSandtheplate.BreakingApartArgument9BreakingApartArgumentnPn2
4
6
8
.
.
n
1
3
5
7
.
.
n-1SSupposeniseven.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.nPn2
4
6
8
.
.
n
1
3
5
7
.
nPnSupposeniseven.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.2
1
SDetail:Thisconstructiononlyworkswhenn>2nPnSupposeniseven.
SconPnSupposenisodd.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.1
3
5
7
.
.
n2
4
6
8
.
.
n-1
SnPnSupposenisodd.
SconnPn1
3
2
SDetail:Thisconstructiononlyworkswhenn>3Supposenisodd.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.nPnSDetail:ThisconstructinPn
2n–3forn≥3BringToTopiswithinafactoroftwoofoptimal!nPn2n–3forn≥3BrinStartingfromANYstackwecangettothesortedstackusingnomorethanPnflips.nPn
2n–3forn≥3StartingfromANYstackwecanFromANYstacktosortedstackin≤Pn.Reversethesequencesweusetosort.FromsortedstacktoANYstackin≤Pn?
((()))FromANYstacktosortedstackHence,
FromANYstacktoANYstackin≤2Pn.
FromANYstacktosortedstackin≤Pn.FromsortedstacktoANYstackin≤Pn?
Hence,
FromANYstacktoANYFromANYstacktoANYstackin≤2Pn.
Canyoufindafasterwaythan2PnflipstogofromANYtoANY?
((()))FromANYstacktoANYstackinFromANYStackStoANYstackTin≤PnRenamethepancakesinStobe1,2,3,..,n.RewriteTusingthenewnamingschemethatyouusedforS.
Twillbesomelist:p(1),p(2),..,p(n).
Thesequenceofflipsthatbringsthesortedstacktop(1),p(2),..,p(n)willbringStoT.S:
4,3,5,1,2T:
5,2,4,3,11,2,3,4,53,5,1,2,4FromANYStackStoANYstackTheKnownPancakeNumbers1
2
3
4
5
6
7
8
9
10
11
12
130
1
3
4
5
nPn7
8
9
10
11
13
14
15TheKnownPancakeNumbers1
2
3P14IsUnknown14!Orderingsof14pancakes.14!=87,178,291,200P14IsUnknown14!OrderingsofIsThisReallyComputerScience?IsThisReallyComputerSciencPosedinAmer.Math.Monthly82(1)(1975),“HarryDweighter”a.k.a.JacobGoodmanPosedinAmer.Math.Monthly8(17/16)nPn
(5n+5)/3WilliamGates&ChristosPapadimitriouBoundsForSortingByPrefixReversal.DiscreteMathematics,vol27,pp47-57,1979.(17/16)nPn(5n+5)/3Willia(15/14)nPn
(5n+5)/3H.Heydari&
H.I.SudboroughOntheDiameterofthePancakeNetwork.JournalofAlgorithms,vol25,pp67-94,1997.(15/14)nPn(5n+5)/3H.HeyPermutationAnyparticularorderingofallnelementsofannelementsetS,iscalledapermutationonthesetS.Example:S={1,2,3,4,5}Onepossiblepermutation:532415*4*3*2*1=120possiblepermutationsonSPermutationAnyparticularordePermutationAnyparticularorderingofallnelementsofannelementsetS,iscalledapermutationonthesetS.Eachdifferentstackofnpancakesisoneofthepermutationson[1..n].PermutationAnyparticularordeRepresentingAPermutationWehavemanywaystospecifyapermutationonS.
Herearetwomethods:Welistasequenceofalltheelementsof[1..n],eachonewrittenexactlyonce.
Ex:645213WegiveafunctiononSsuchthat
(1)(2)(3)..(n)isasequencethat
lists[1..n],eachoneexactlyonce.
Ex:(1)=6(2)=4(3)=5(4)=2(4)=1(6)=3RepresentingAPermutationWehAPermutationisaNOUNAnorderingSofastackofpancakesisapermutation.APermutationisaNOUNAnordeAPermutationisaNOUN.
ApermutationcanalsobeaVERB.AnorderingSofastackofpancakesisapermutation.WecanpermuteStoobtainanewstackS’.Permutealsomeanstorearrangesoastoobtainapermutationoftheoriginal.APermutationisaNOUN.
AperPermuteAPermutation.IstartwithapermutationSofpancakes.
Icontinuetouseaflipoperationtopermutemycurrentpermutation,soastoobtainthesortedpermutation.PermuteAPermutation.IstartTherearen!=1*2*3*4*…*npermutationsonnelements.Easyproofinthefirstcountinglecture.Therearen!=1*2*3*4*…*nperPancakeNetwork:
DefinitionForn!NodesForeachnode,assignitthenameofoneofthen!stacksofnpancakes.Putawirebetweentwonodesiftheyareoneflipapart.PancakeNetwork:
DefinitionFoNetworkForn=3123321213312231132NetworkForn=3123321213312231NetworkForn=4NetworkForn=4PancakeNetwork:
MessageRoutingDelayWhatisthemaximumdistancebetweentwonodesinthenetwork?PnPancakeNetwork:
MessageRoutiPancakeNetwork:
ReliabilityIfupton-2nodesgethitbylightningthenetworkremainsconnected,eventhougheachnodeisconnectedtoonlyn-1othernodes.ThePancakeNetworkisoptimallyreliableforitsnumberofedgesandnodes.PancakeNetwork:
ReliabilityIfMutationDistanceMutationDistanceHighLevelPointComputerScienceisnotmerelyaboutcomputersandprogramming,itisaboutmathematicallymodelingourworld,andaboutfindingbetterandbetterwaystosolveproblems.Thislectureisamicrocosmofthisexercise.HighLevelPointOne“Simple”ProblemAhostofproblemsandapplicationsatthefrontiersofscienceOne“Simple”ProblemAhostofStudyBeePleasereadthe
coursedocumentcarefully.Youmusthandina
signedcheating
policypage.StudyBeeStudyBeeDefinitionsof: nthpancakenumber lowerbound upperbound permutationProofof: ANYtoANYin≤PnImportantTechnique: BracketingStudyBeeReferencesBillGates&ChristosPapadimitriou:BoundsForSortingByPrefixReversal.DiscreteMathematics,vol27,pp47-57,1979.H.Heydari&H.I.Sudborough:OntheDiameterofhePancakeNetwork.JournalofAlgorithms,vol25,pp67-94,1997ReferencesPancakesWithAProblem!GreatTheoreticalIdeasInComputerScienceAnupamGuptaCS15-251Fall2O05Lecture1Aug29th,2OO5CarnegieMellonUniversityPancakesWithAProblem!GreatCourseStaffProfs: AnupamGupta JohnLaffertyTAs: KanatTangwongsan YinmengZhang /~15251Pleasecheckthewebpagesregularly!CourseStaffPleasefeelfreetoaskquestions!
((()))Pleasefeelfree((()))CourseDocument
Youmustreadthiscarefully.Gradingformulaforthecourse.40%homework
5%in-classquizzes25%in-recitationtests30%finalSevenpointsadaylatepenalty.Collaboration/CheatingPolicyYoumayNOTsharewrittenwork.Wereusehomeworkproblems.CourseDocument
YoumustreadPancakesWithAProblem!GreatTheoreticalIdeasInComputerScienceAnupamGuptaCS15-251Fall2O05Lecture1Aug29th,2OO5CarnegieMellonUniversityPancakesWithAProblem!GreatThechefatourplaceissloppy,andwhenhepreparesastackofpancakestheycomeoutalldifferentsizes.Therefore,whenIdeliverthemtoacustomer,onthewaytothetableIrearrangethem(sothatthesmallestwindsupontop,andsoon,downtothelargestatthebottom)
Idothisbygrabbingseveralfromthetopandflippingthemover,repeatingthis(varyingthenumberIflip)asmanytimesasnecessary.ThechefatourplaceissloppDevelopingANotation:
TurningpancakesintonumbersDevelopingANotation:
TurningDevelopingANotation:
TurningpancakesintonumbersDevelopingANotation:
TurningDevelopingANotation:
Turningpancakesintonumbers23451DevelopingANotation:
TurningDevelopingANotation:
Turningpancakesintonumbers52341DevelopingANotation:
TurningHowdowesortthisstack?
Howmanyflipsdoweneed?5
2
3
4
1Howdowesortthisstack?
How4FlipsAreSufficient12345523414321523415143254FlipsAreSufficient12345523AlgebraicRepresentation52341X=Thesmallestnumber
offlipsrequiredtosort:?X?UpperBoundLowerBoundAlgebraicRepresentation52341XAlgebraicRepresentation52341X=Thesmallestnumber
offlipsrequiredtosort:?X4UpperBoundLowerBoundAlgebraicRepresentation52341X4Flipsarenecessaryinthiscase523414132514325Flip1hastoput5onbottomFlip2mustbring4totop.4FlipsarenecessaryinthisUpperBoundLowerBound4X4X=4UpperBoundLowerBound45thPancakeNumberThenumberofflipsrequiredtosorttheworstcasestackof5pancakes.?P5
?UpperBoundLowerBoundP5=5thPancakeNumberThenumbero5thPancakeNumberThenumberofflipsrequiredtosorttheworstcasestackof5pancakes.4P5
?UpperBoundLowerBoundP5=5thPancakeNumberThenumbero..........The5thPancakeNumber:
TheMAXoftheX’s120119932X1X2X3X119X120523414..........The5thPanThe5thPancakeNumber:
TheMAXoftheX’s1201993..........X1X2X3X119X1205234141234554321The5thPancakeNumber:
TheM1201993..........X1X2X3X119X1205234141234554321P5=
MAXovers2stacksof5
ofMIN#offlipstosorts
1201993..........X1X2Pn=
MAXovers2stacksofnpancakes
ofMIN#offlipstosorts
Pn=
Thenumberofflipsrequiredtosorttheworst-casestackofnpancakes.Pn=
MAXovers2stacksofnPn=
MAXovers2stacksofnpancakes
ofMIN#offlipstosorts
Pn=
Thenumberofflipsrequiredtosortaworst-casestackofnpancakes.Pn=
MAXovers2stacksofnBeCool.LearnMath-speak.Pn=
Thenumberofflipsrequiredtosorta
worst-casestackofnpancakes.Pn=
ThenumberofflipsrequWhatisPnforsmalln?Canyoudon=0,1,2,3?WhatisPnforsmalln?CanyouInitialValuesOfPnn0123Pn0013InitialValuesOfPnn0123Pn001P3=31
3
2requires3Flips,henceP3≥3.ANYstackof3canbedonebygettingthebigonetothebottom(≤2flips),andthenusing≤1extrafliptohandlethetoptwo.Hence,P3=3.P3=31
3
2requires3nthPancakeNumberThenumberofflipsrequiredtosortaworstcasestackofnpancakes.?Pn
?UpperBoundLowerBoundPn=nthPancakeNumberThenumberoBracketing:
WhatarethebestlowerandupperboundsthatIcanprove?≤f(x)≤[]Bracketing:
Whatarethebest?Pn
?TakeafewminutestotryandproveupperandlowerboundsonPn,
forn>3.?Pn?TakeafewminutestBringbiggesttotop.
Placeitonbottom.Bringnextlargesttotop.
Placesecondfrombottom.Andsoon…Bring-to-topMethodBringbiggesttotop.
PlaceiUpperBoundOnPn:
BringToTopMethodFornPancakesIfn=1,noworkrequired-wearedone!Otherwise,flippancakentotopand
thenflipittopositionn.Nowuse:BringToTopMethod
Forn-1PancakesTotalCost:atmost2(n-1)=2n–2flips.UpperBoundOnPn:
BringToToBetterUpperBoundOnPn:
BringToTopMethodFornPancakesIfn=2,atmostoneflipandwearedone.Otherwise,flippancakentotopand
thenflipittopositionn.Nowuse:BringToTopMethod
Forn-1PancakesTotalCost:atmost2(n-2)+1=2n–3flips.BetterUpperBoundOnPn:
BrinBringtotopnotalwaysoptimalforaparticularstackBring-to–toptakes5flips,butwecandoin4flips3214552341231454132514325Bringtotopnotalwaysoptima?Pn
2n-3WhatotherboundscanyouproveonPn??Pn2n-3Whatotherbounds9
16SupposeastackScontainsapairofadjacentpancakesthatwillnotbeadjacentinthesortedstack.AnysequenceofflipsthatsortsstackSmustinvolveoneflipthatinsertsthespatulabetweenthatpairandbreaksthemapart.BreakingApartArgument9BreakingApartArgument9
16SupposeastackScontainsapairofadjacentpancakesthatwillnotbeadjacentinthesortedstack.AnysequenceofflipsthatsortsstackSmustinvolveoneflipthatinsertsthespatulabetweenthatpairandbreaksthemapart.Furthermore,thissameprincipleistrueofthe“pair”formedbythebottompancakeofSandtheplate.BreakingApartArgument9BreakingApartArgumentnPn2
4
6
8
.
.
n
1
3
5
7
.
.
n-1SSupposeniseven.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.nPn2
4
6
8
.
.
n
1
3
5
7
.
nPnSupposeniseven.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.2
1
SDetail:Thisconstructiononlyworkswhenn>2nPnSupposeniseven.
SconPnSupposenisodd.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.1
3
5
7
.
.
n2
4
6
8
.
.
n-1
SnPnSupposenisodd.
SconnPn1
3
2
SDetail:Thisconstructiononlyworkswhenn>3Supposenisodd.
ScontainsnpairsthatwillneedtobebrokenapartduringanysequencethatsortsstackS.nPnSDetail:ThisconstructinPn
2n–3forn≥3BringToTopiswithinafactoroftwoofoptimal!nPn2n–3forn≥3BrinStartingfromANYstackwecangettothesortedstackusingnomorethanPnflips.nPn
2n–3forn≥3StartingfromANYstackwecanFromANYstacktosortedstackin≤Pn.Reversethesequencesweusetosort.FromsortedstacktoANYstackin≤Pn?
((()))FromANYstacktosortedstackHence,
FromANYstacktoANYstackin≤2Pn.
FromANYstacktosortedstackin≤Pn.FromsortedstacktoANYstackin≤Pn?
Hence,
FromANYstacktoANYFromANYstacktoANYstackin≤2Pn.
Canyoufindafasterwaythan2PnflipstogofromANYtoANY?
((()))FromANYstacktoANYstackinFromANYStackStoANYstackTin≤PnRenamethepancakesinStobe1,2,3,..,n.RewriteTusingthenewnamingschemethatyouusedforS.
Twillbesomelist:p(1),p(2),..,p(n).
Thesequenceofflipsthatbringsthesortedstacktop(1),p(2),..,p(n)willbringStoT.S:
4,3,5,1,2T:
5,2,4,3,11,2,3,4,53,5,1,2,4FromANYStackStoANYstackTheKnownPancakeNumbers1
2
3
4
5
6
7
8
9
10
11
12
130
1
3
4
5
nPn7
8
9
10
11
13
14
15TheKnownPancakeNumbers1
2
3P14IsUnknown14!Orderingsof14pancakes.14!=87,178,291,200P14IsUnknown14!OrderingsofIsThisReallyComputerScience?IsThisReallyComputerSciencPosedinAmer.Math.Monthly82(1)(1975),“HarryDweighter”a.k.a.JacobGoodmanPosedinAmer.Math.Monthly8(17/16)nPn
(5n+5)/3WilliamGates&ChristosPapadimitriouBoundsForSortingByPrefixReversal.DiscreteMathematics,vol27,pp47-57,1979.(17/16)nPn(5n+5)/3Willia(15/14)nPn
(5n+5)/3H.Heydari&
H.I.SudboroughOntheDiameterofthePancakeNetwork.JournalofAlgorithms,vol25,pp67-94,1997.(15/14)nPn(5n+5)/3H.HeyPermutationAnyparticularorderingofallnelementsofannelementsetS,iscalledapermutationonthesetS.Example:S={1,2,3,4,5}Onepossiblepermutation:532415*4*3*2*1=120possiblepermutationsonSPermutationAnyparticularordePermutationAnyparticularorderingofallnelementsofannelementsetS,iscalledapermutationonthesetS.Eachdifferentstackofnpancakesisoneofthepermutationson[1..n].PermutationAnyparticularordeRepresentingAPermutationWehavemanywaystospecifyapermutationonS.
Herearetwomethods:Welistasequenceofalltheelementsof[1..n],eachonewrittenexactlyonce.
Ex:645213WegiveafunctiononSsuchthat
(1)(2)(3)..(n)isasequencethat
lists[1..n],eachoneexactlyonce.
Ex:(1)=6
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