Lecture12Bellman-Ford,Floyd-Warshall,andDynamic

Programming!

Announcements

• HW5dueFriday• Midtermshavebeengraded!• Pickupyourexamafterclass.• Average:84,Median:87• Max:100(x4)

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Lasttime

• Dijkstra’salgorithm!• Solvessingle-sourceshortestpathinweightedgraphs.

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Today

• Bellman-Fordalgorithm• Anothersingle-sourceshortestpathalgorithm

• Thisisanexampleofdynamicprogramming• We’llseewhatthatmeans

• Floyd-Warshall algorithm• An“all-pairs”shortestpathalgorithm• Anotherexampleofdynamicprogramming

Recall• Aweighteddirectedgraph:

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• Weights onedgesrepresentcosts.

• Thecostofapathisthesumoftheweightsalongthatpath.

• Ashortestpathfromstotisadirectedpathfromstotwiththesmallestcost.

• Thesingle-sourceshortestpath problemistofindtheshortestpathfromstovforallvinthegraph.

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Thisisapathfromstotofcost22.

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Thisisapathfromstotofcost10.Itistheshortestpathfromstot.

OnedrawbacktoDijkstra• Mightnotworkwithnegativeedgeweights• Onyourhomework!

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Whywouldweeverhavenegativeweights?• Negativecostsmight

meanbenefits.• eg,itcostsme-$2

whenIget$2.

Bellman-FordAlgorithm

• Slower(butarguablysimpler)thanDijkstra’salgorithm.• Workswithnegativeedgeweights.

Bellman-FordAlgorithm

• Wekeep*anarrayd(k) oflengthnforeachk=0,1,…,n-1.

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*Wewon’tactuallystoreallthese,butlet’spretendwedofornow.

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit,forallbinV.

suvt

d(0)

suvt

d(1)

suvt

d(2)

suvt

d(3)

Formally,wewillmaintaintheloopinvariant:

• Forexample,this istheshortestpathfromstotwithatmosttwoedgesinit.

• Butit’snottheshortestpathfromstot(withanynumberofedges).• That’sthisone.

Bellman-FordAlgorithm

• Wekeep*anarrayd(k) oflengthnforeachk=0,1,…,n-1.

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*Wewon’tactuallystoreallthese,butlet’spretendwedofornow.

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit,forallbinV.

0

∞

∞∞

0 ∞ ∞ ∞

suvt

d(0)

suvt

d(1)

suvt

d(2)

suvt

d(3)

Formally,wewillmaintaintheloopinvariant:

Nowupdate!

• Wewillusethetabled(0) tofillind(1)

• Thenused(1) tofillind(2)

• …• Thenused(k-1) tofillind(k)

• ...• Thenused(n-2) tofillind(n-1)

Thiseventuallygivesuswhatwewant:• d(k)[a]istheshortestpathfromstoawithatmostkedges.• Eventuallywe’llgetalltheshortestpaths…

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Whilemaintaining:

Howdowegetd(k)[b]fromd(k-1)?• Twocases: d(k)[b]isthecostoftheshortestpath

fromstobwithatmostkedgesinit.

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x2

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b102

22

Case1:theshortestpathfromstobwithatmostkedgesactuallyhasatmostk-1edges.

Case2:theshortestpathfromstobwithatmostkedgesreallyhaskedges.

d(k)[b]=d(k-1)[b]

d(k)[b]=d(k-1)[a]+w(a,b)forsomea...

sayk=3

sayk=3

Wanttomaintain:

d(k)[b]=mina {d(k-1)[a]+w(a,b)}

Bellman-FordAlgorithm*

• Bellman-Ford*(G,s):• Initialize d(k) fork=0,…,n-1• d(0)[v]=∞ forallvotherthans• d(0)[s]=0.• Fork=1,…,n-1:

• ForbinV:• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

• Returnd(n-1)

Ifwesetd(k)[b]tobetheminimumoftheprevioustwocases,thenwemaintaintheloopinvariantthat:

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Thisminimumisoverallasothat(a,b)isinE

Bellman-FordAlgorithm*Example

• Fork=1,…,n-1:• ForbinV:

• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

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0 ∞ ∞ ∞

suvt

d(0)

suvt

d(1)

suvt

d(2)

suvt

d(3)

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

0

∞

∞∞

Bellman-FordAlgorithm*Example

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0 ∞ ∞ ∞

suvt

d(0)

0 2 5 ∞

suvt

d(1)

suvt

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suvt

d(3)

0

2

5∞

• Fork=1,…,n-1:• ForbinV:

• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Bellman-FordAlgorithm*Example

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0 ∞ ∞ ∞

suvt

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• Fork=1,…,n-1:• ForbinV:

• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Bellman-FordAlgorithm*Example

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0 ∞ ∞ ∞

suvt

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suvt

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suvt

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• Fork=1,…,n-1:• ForbinV:

• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Bellman-FordAlgorithm*Example

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0 ∞ ∞ ∞

suvt

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0 2 5 ∞

suvt

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SANITYCHECK:• Theshortestpathwith1edgefromstothascost∞. (thereisnosuchpath).• Theshortestpathwith2edgesfromstothascost3.(s-v-t)• Theshortestpathwith3edgesfromstothascost2.(s-u-v-t)

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

Andthisoneistheshortestpath!!!

Howdoweactuallyimplementthis?(Thisiswhatthe*onallthepreviousslideswasfor).

• Don’tactuallykeepallthearraysd(k) around.• Justkeeptwoofthematatime,that’sallweneed.

• Runningtime:O(mn)• That’sworsethanDijkstra,butBFcanhandlenegativeedgeweights.

• Spacecomplexity:• Weneedspacetostorethegraphandtwoarraysofsizen.

*WARNING: ThisisslightlydifferentfromtheversionofBellman-FordinCLRS.Butwewillstickwithwhatwejustsawforpedagogicalreasons.SeeLectureNotes11.5(listedonthewebpageintheLecture12box)fornotesontheanalysisoftheslightlydifferentCLRSversion.

Bellman-FordAlgorithm*

• Bellman-Ford*(G,s):• Initialize d(k) fork=0,…,n-1• d(0)[v]=∞ forallvotherthans• d(0)[s]=0.• Fork=1,…,n-1:

• ForbinV:• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

• Returnd(n-1)

Whydoesitwork?

• First,we’vebeenassertingthat:

• Technically,thisrequiresproof!• We’vebasicallyalreadyseentheproof!• Itfollowsfrominductionwiththeinductivehypothesis

d(n-1)[b]isthecostoftheshortestpathfromstobwithatmostn-1edgesinit.

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.Workoutthedetailsofthisproof

onyourown!Tohelpyou,there’sanoutlineonthenextslide.(Whichwe’llskipnow).

Sketchofproof[skipthisinlecture]thatthisthingwe’vebeenassertingisreallytrue

• Inductivehypothesis:

• Basecase:

• Inductivestep:

• Conclusion:

d(k)[b]isthecostoftheshortestpathfromstobwithatmostkedgesinit.

0 ∞ ∞ ∞Fork=0:

• d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

• Ineithercase,wemakethecorrectupdate.

Case1:theshortestpathfromstobehas<kedges

Case2:theshortestpathfromstoboflengthatmostkedgeshas

exactlykedges

d(n-1)[b]isthecostoftheshortestpathfromstobwithatmostn-1edgesinit.

Whenk=n-1,theinductivehypothesisreads:

Isthistheconclusionwewant?

• WestillneedtoprovethatthisimpliesBF*iscorrect.• Wereturnd(n-1)• Needtoshowd(n-1)[a]=distance(s,a).

• Enoughtoshow:

d(n-1)[b]isthecostoftheshortestpathfromstobwithatmostn-1edgesinit.

Shortestpathwithatmostn-1edges

Shortestpathwithanynumberofedges

DANGER!

• Ifthegraphhasanegativecycle,thismightnotbetrue.• Ifthereisanegativecycle,theremaynotbeashortestpathbetweentwovertices!

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Shortestpathwithatmostn-1

edges

Shortestpathwithanynumberofedges

Anegativecycleisadirectedcyclewithnegativetotalcost

Butifthereisnonegativecycle

• Thennotonlyarethereshortestpaths,butactuallythere’salwaysasimple shortestpath.

• Asimplepathinagraphwithnverticeshasatmostn-1edgesinit.

“Simple”meansthatthepathhasnocyclesinit.

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Can’taddanotheredgewithoutmakingacycle!

Thiscycleisn’thelping.Justgetridofit.

Let’sgoafteranewconclusion.

• Theorem:• TheBellman-FordAlgorithm*iscorrectaslongasGhasnonegativecycles.

*WewillprovethisforourversionofBellman-Ford.SeeNotes11.5orCLRSforCLRSversion.

Proof• Byinduction,

• Iftherearenonegativecycles,

• ThisisbecausetheshortestpathisWLOGsimple,andallsimplepathshaveatmostn-1edges.

• Sothethingwereturnisequaltothethingwewanttoreturn.

d(n-1)[b]isthecostoftheshortestpathfromstobwithatmostn-1edgesinit.

Shortestpathwithatmostn-1

edges

Shortestpathwithanynumberofedges

Sothatproves:

• Theorem:• TheBellman-FordAlgorithm*iscorrectaslongasGhasnonegativecycles.

• Further,ifGhasanegativecycle,Bellman-Fordcandetectthat.• (SeeNotes11.5)

Whathavewelearned?

• TheBellman-FordalgorithmisslowerthanDijkstra:• O(mn)time

• Butitworkswithnegativeedgesweights.• You’llseehowDijkstradoeswithnegativeedgeweightsinHW5.

• Itdoesn’tworkwithnegativecycles,butinthatcaseshortestpathsdon’tevenmakesense.

Bellman-Fordisalsousedinpractice.• eg,RoutingInformationProtocol(RIP)usessomethinglikeBellman-Ford.• Olderprotocol,notusedasmuchanymore.

• Eachrouterkeepsatable ofdistancestoeveryotherrouter.• PeriodicallywedoaBellman-Fordupdate.• Thisalsomeansthatiftherearechangesinthenetwork,thiswillpropagate.(maybeslowly…)

Destination Costtogetthere

Sendtowhom?

172.16.1.0 34 172.16.1.1

10.20.40.1 10 192.168.1.2

10.155.120.1 9 10.13.50.0

Thiswasanexampleof…

Whatisdynamicprogramming?

• Itisanalgorithmdesignparadigm• likedivide-and-conquerisanalgorithmdesignparadigm.

• Usuallyitisforsolvingoptimizationproblems• eg,shortestpath

Elementsofdynamicprogramming

• Bigproblemsbreakupintolittleproblems.• eg,Shortestpathwithatmostkedges.

• Theoptimalsolutionofaproblemcanbeexpressedintermsofoptimalsolutionsofsmallersub-problems.• eg,d(k)[b]←min{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

Wecallthis“optimalsub-structure”

ElementsofdynamicprogrammingII

• Thesub-problemsoverlapalot.• eg,Lotsofdifferententriesofd(k) askford(k-1)[a].• Thismeansthatwecansavetimebysolvingasub-problemjustonceandstoringtheanswer.

Wecallthis“overlappingsub-problems”

ElementsofdynamicprogrammingIII• Optimalsubstructure.• Optimalsolutionstosub-problemsaresub-solutionstotheoptimalsolutionoftheoriginalproblem.

• Overlappingsubproblems.• Thesubproblems showupagainandagain

• Usingtheseproperties,wecandesignadynamicprogrammingalgorithm:• Keepatableofsolutionstothesmallerproblems.• Usethesolutionsinthetabletosolvebiggerproblems.• Attheendwecanuseinformationwecollectedalongthewaytofindtheoptimalsolution.• eg, recovertheshortestpath(notjustitscost).

Twowaystothinkaboutand/orimplement DPalgorithms

• Topdown

•Bottomup

This picture isn’t hugely relevant but I like it.

Bottomupapproach

• Whatwejustsaw.• Solvethesmallproblemsfirst• fillind(0)

• Thenbiggerproblems• fillind(1)

• …• Thenbiggerproblems• fillind(n-2)

• Thenfinallysolvetherealproblem.• fillind(n-1)

Topdownapproach

• Thinkofitlikearecursivealgorithm.• Tosolvethebigproblem:• Recurse tosolvesmallerproblems

• Thoserecurse tosolvesmallerproblems• etc..

• Thedifferencefromdivideandconquer:• Memo-ization• Keeptrackofwhatsmallproblemsyou’vealreadysolvedtopreventre-solvingthesameproblemtwice.

Example:top-down**versionofBF*

• Bellman-Ford*(G,s):• Initializeabunchofemptytablesd(k) fork=0,…,n-1,• Fillind(0)

• for binV:• BF*_helper(G,s,b,n-1)

• BF*_helper(G,s,b,k):• Foreachasothat(a,b)inE,andalsofora=b:

• If d(k-1)[a]isnotalreadyinthetable:• d(k-1)[a]=BF*_helper(G,s,a,k-1)

• returnmin{d(k-1)[b],mina {d(k-1)[a]+weight(a,b)}}

*NottheactualBellman-Fordalgorithm;wedon’twanttokeepallthesetablesaround**ProbablynotthebestwaytothinkaboutBellman-Ford:thisisforDPpedagogyonly!

Theactualpseudocodehereisn’timportant,Ijustwanttotalkaboutthestructureofit.

Visualizationtop-downapproach

d(n-1)[u] d(n-1)[v]

d(n-2)[a]d(n-2)[a]d(n-2)[y]d(n-2)[x] d(n-2)[x]

d(n-3)[v]d(n-3)[t]d(n-3)[z] d(n-3)[v]d(n-3)[x]

Thisisareallybigrecursiontree!Naively,nlayers,soatleast2n

time!

Visualizationtop-downapproach

d(n-1)[u] d(n-1)[v]

d(n-2)[a]d(n-2)[y]d(n-2)[x]

d(n-3)[v]d(n-3)[t]d(n-3)[z]d(n-3)[x]

Nowit’samuchsmaller

“recursionDAG!”

d(n-2)[z]

Whathavewelearned?

• Paradigminalgorithmdesign.• Usefulwhenthere’soptimalsubstructure:

• optimalsolutionstoabigproblembreakupintooptimalsub-solutionsofsubproblems.

• Usefulwhenthereareoverlappingsubproblems:• Usememo-ization (aka,putitinatable)topreventrepeatedwork.

• Canbeimplementedbottom-up ortop-down.• It’safancynameforaprettycommon-senseidea:

• Don’tduplicateworkifyoudon’thaveto!

Why“dynamicprogramming”?

• Programming referstofindingtheoptimal“program.”• asin,ashortestrouteisaplan akaaprogram.

• Dynamic referstothefactthatit’smulti-stage.• Butalsoit’sjustafancy-soundingname.

Manipulatingcomputercodeinanactionmovie?

Why“dynamicprogramming”?

• RichardBellmaninventedthenameinthe1950’s.• Atthetime,hewasworkingfortheRANDCorporation,whichwasbasicallyworkingfortheAirForce,andgovernmentprojectsneededflashynamestogetfunded.• FromBellman’sautobiography:• “It’s impossible to use the word, dynamic, in the

pejorative sense…I thought dynamic programming was a good name. It was something not even a Congressman could object to.”

Anotherexample

• Floyd-Warshall Algorithm• ThisisanalgorithmforAll-PairsShortestPaths(APSP)• Thatis,IwanttoknowtheshortestpathfromutovforALLpairsu,v ofverticesinthegraph.• Notjustfromaspecialsinglesources.

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Source

Destination

Anotherexample

• Floyd-Warshall Algorithm• ThisisanalgorithmforAll-PairsShortestPaths(APSP)• Thatis,IwanttoknowtheshortestpathfromutovforALLpairsu,v ofverticesinthegraph.• Notjustfromaspecialsinglesources.

• Naïvesolution(ifwewanttohandlenegativeedgeweights):• ForallsinG:

• RunBellman-FordonGstartingats.• TimeO(n⋅nm)=O(n2m),

• maybeasbadasn4 ifm=n2

Optimalsubstructure

k-1

2

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3

kk+1

uv

n

Labelthevertices1,2,…,n(Weomitedgesinthe

picturebelow).

LetD(k-1)[u,v]bethesolutiontothissub-problem.

Thisistheshortestpathfromutovthroughtheblueset.Ithaslength

D(k-1)[u,v]

Sub-problem:Forallpairs,u,v,findthecostoftheshortestpathfromutov,sothatalltheinternalverticesonthatpatharein{1,…,k-1}.

Optimalsubstructure

k-1

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n

Sub-problem:Forallpairs,u,v,findthecostoftheshortestpathfromutov,sothatalltheinternalverticesonthatpatharein{1,…,k-1}.

Labelthevertices1,2,…,n(Weomitedgesinthe

picturebelow).

LetD(k-1)[u,v]bethesolutiontothissub-problem.

Thisistheshortestpathfromutovthroughtheblueset.Ithaslength

D(k-1)[u,v]

Question:HowcanwefindD(k)[u,v]usingD(k-1)?

HowcanwefindD(k)[u,v]usingD(k-1)?

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D(k)[u,v]isthecostoftheshortestpathfromutovsothatallinternalverticesonthatpatharein{1,…,k}.

HowcanwefindD(k)[u,v]usingD(k-1)?

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D(k)[u,v]isthecostoftheshortestpathfromutovsothatallinternalverticesonthatpatharein{1,…,k}.

Case1:wedon’tneedvertexk.

D(k)[u,v]=D(k-1)[u,v]

HowcanwefindD(k)[u,v]usingD(k-1)?

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D(k)[u,v]isthecostoftheshortestpathfromutovsothatallinternalverticesonthatpatharein{1,…,k}.

Case2:weneedvertexk.

Case2continued

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uv

n

• Supposetherearenonegativecycles.• ThenWLOGtheshortestpathfrom

utovthrough{1,…,k}issimple.

• Ifthatpath passesthroughk,itmustlooklikethis:

• Thispathistheshortestpathfromutokthrough{1,…,k-1}.• sub-pathsofshortestpathsare

shortestpaths• Sameforthispath.

Case2:weneedvertexk.

D(k)[u,v]=D(k-1)[u,k]+D(k-1)[k,v]

HowcanwefindD(k)[u,v]usingD(k-1)?

• D(k)[u,v]=min{D(k-1)[u,v],D(k-1)[u,k] +D(k-1)[k,v]}

• Optimalsubstructure:• Wecansolvethebigproblemusingsmallerproblems.

• Overlappingsub-problems:• D(k-1)[k,v]canbeusedtohelpcomputeD(k)[u,v]forlotsofdifferentu’s.

Case1:Costofshortestpath

through{1,…,k-1}

Case2:Costofshortestpathfromutokandthenfromktov

through{1,…,k-1}

HowcanwefindD(k)[u,v]usingD(k-1)?

• D(k)[u,v]=min{D(k-1)[u,v],D(k-1)[u,k] +D(k-1)[k,v]}

• Usingour paradigm,thisimmediatelygivesusanalgorithm!

Case1:Costofshortestpath

through{1,…,k-1}

Case2:Costofshortestpathfromutokandthenfromktov

through{1,…,k-1}

Floyd-Warshall algorithm

• Initializen-by-narraysD(k)fork=0,…,n• D(k)[u,u]=0forallu,forallk• D(k)[u,v]=∞ forallu≠ v,forallk• D(0)[u,v]=weight(u,v)forall(u,v)inE.

• For k=1,…,n:• For pairsu,v inV2:• D(k)[u,v]=min{D(k-1)[u,v],D(k-1)[u,k] +D(k-1)[k,v]}

• Return D(n)

Thebasecasechecksout:the

onlypaththroughzerootherverticesareedgesdirectly

fromutov.

Thisisabottom-up algorithm.

We’vebasicallyjustshown• Theorem:

IftherearenonegativecyclesinaweighteddirectedgraphG,thentheFloyd-Warshall algorithm,runningonG,returnsamatrixD(n) sothat:

D(n)[u,v]=distancebetweenuandvinG.

• Runningtime:O(n3)• BetterthanrunningBFntimes!• NotreallybetterthanrunningDijkstrantimes.

• Butit’ssimplertoimplementandhandlesnegativeweights.

• Storage:• Enoughtoholdtwo n-by-narrays,andtheoriginalgraph.

Workoutthedetailsoftheproof!(OrseeLecture

Notes12forafewmoredetails).

AswithBellman-Ford,wedon’treallyneedtostoreallnoftheD(k).

Whatifthereare negativecycles?

• JustlikeBellman-Ford,Floyd-Warshall candetectnegativecycles.• Ifthereisanegativecycle,thenthereisapathfromvtovthatgoesthroughallnverticesthathascost<0.• That’sjustthedefinitionofanegativecycle.

• SoD(n)[v,v]<0.• Socheckforthatattheend.• ifthereissuchav,returnnegative cycle.

Whathavewelearned?

• TheFloyd-Warshall algorithmisanotherexampleofdynamicprogramming.

• ItcomputesAllPairsShortestPathsinadirectedweightedgraphintimeO(n3).

AnotherExample?

• Longestsimplepath(sayalledgeweightsare1):

b

a

Whatisthelongestsimplepathfromstot?

s

t

Thisisanoptimizationproblem…

• CanweuseDynamicProgramming?• OptimalSubstructure?• Longestpathfromstot=longestpathfromstoa

+longestpathfromatot?

b

as

t

NOPE!

Thisdoesn’twork

• Thesubproblems wecameupwitharen’t independent:• Oncewe’vechosenthelongestpathfromatot

• whichusesb,• ourlongestpathfromstoashouldn’tbeallowedtouseb

• sinceb wasalreadyused.

b

as

t

Whatwentwrong?

• Actually,thelongestsimplepathproblemisNP-complete.• Wedon’tknowofanypolynomial-timealgorithmsforit,DPorotherwise!

Recap

• Twomoreshortest-pathalgorithms:• Bellman-Fordforsingle-sourceshortestpath• Floyd-Warshall forall-pairsshortestpath

• Dynamicprogramming!• Thisisafancynamefor:

• Breakupanoptimizationproblemintosmallerproblems• Theoptimalsolutionstothesub-problemsshouldbesub-solutionstotheoriginalproblem.

• Buildtheoptimalsolutioniterativelybyfillinginatableofsub-solutions.

• Takeadvantageofoverlappingsub-problems!

Nexttime

• Moreexamplesofdynamicprogramming!

Wewillstopbulletswithouraction-packedcodingskills,andalsomaybefindlongestcommonsubsequences.