Objecves Review: Maximum Flow Problemsprenkle/cs211/winter16/lectures/29-networkflow.pdf · 4/4/16...
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4/4/16 1 Objec&ves • Network Flow Ø Mo&va&on Ø Max flow Ø Min cut Apr 4, 2016 1 CSCI211 - Sprenkle Review: Maximum Flow Problem • Make network most efficient Ø Use most of available capacity Apr 4, 2016 CSCI211 - Sprenkle 2 10 9 9 14 4 10 4 8 9 1 0 0 0 14 capacity flow s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 10 10 15 4 4 0 Value = 28 Goal: Find s-t flow of maximum value Review: Residual Graph: G f • Original edge: e = (u, v) ∈ E Ø Flow f(e), capacity c(e) • Residual edge Ø e = (u, v) w/ capacity c(e) - f(e) Ø e R = (v, u) with capacity f(e) • To undo flow • Residual graph:G f = (V, E f ) Ø Residual edges with posi(ve residual capacity Ø E f = {e : f(e) < c(e)} ∪ {e R : f(e) > 0} Apr 4, 2016 CSCI211 - Sprenkle 3 u v 11 residual capacity 6 residual capacity Forward edges Backward edges u v 17 6 capacity flow Augmen&ng Path Algorithm Apr 4, 2016 CSCI211 - Sprenkle 4 Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow G f = residual graph while there exists augmenting path P f = Augment(f, c, P) # change the flow update G f # build a new residual graph return f Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(e R ) = f(e) - b # forward edge, ê flow return f c=capacity Ford-Fulkerson Algorithm Apr 4, 2016 CSCI211 - Sprenkle 5 s 2 3 4 5 t 10 10 9 8 4 10 10 6 2 0 0 0 0 0 0 0 0 G: Flow value = 0 0 flow capacity Ford-Fulkerson Algorithm Apr 4, 2016 CSCI211 - Sprenkle 6 s 2 3 4 5 t 10 10 9 8 4 10 10 6 2 0 0 0 0 0 0 0 0 G: Flow value = 0 0 flow What does the residual graph look like? capacity
Transcript of Objecves Review: Maximum Flow Problemsprenkle/cs211/winter16/lectures/29-networkflow.pdf · 4/4/16...
4/4/16
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Objec&ves• NetworkFlow
Ø Mo&va&onØ MaxflowØ Mincut
Apr4,2016 1CSCI211-Sprenkle
Review:MaximumFlowProblem• Makenetworkmostefficient
Ø Usemostofavailablecapacity
Apr4,2016 CSCI211-Sprenkle 2
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capacityflow
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Value = 28
Goal:Finds-tflowofmaximumvalue
Review:ResidualGraph:Gf• Originaledge:e=(u,v)∈E
Ø Flowf(e),capacityc(e)• Residualedge
Ø e=(u,v)w/capacityc(e)-f(e)Ø eR=(v,u)withcapacityf(e)
• Toundoflow• Residualgraph:Gf=(V,Ef)
Ø Residualedgeswithposi(veresidualcapacityØ Ef={e:f(e)<c(e)}∪{eR:f(e)>0}
Apr4,2016 CSCI211-Sprenkle 3
u v 11
residual capacity
6
residual capacity
Forward edges Backward edges
u v 17
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capacity
flow
Augmen&ngPathAlgorithm
Apr4,2016 CSCI211-Sprenkle 4
Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow Gf = residual graph
while there exists augmenting path P f = Augment(f, c, P) # change the flow update Gf # build a new residual graph
return f
Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(eR) = f(e) - b # forward edge, ê flow return f
c=capacity
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 5
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flow
capacity
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 6
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What does the residual graph look like?
capacity
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Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 7
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capacity
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 8
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residual capacity
Bottleneck
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Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 9
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Flow value = 8
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 10
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Flow value = 10
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 11
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Flow value = 16
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 12
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Flow value = 18
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Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 13
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Flow value = 19
How do we know we’re done?
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 14
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What is reachable from s
Flow value = 19Cut capacity = 19
AnalyzingAugmen&ngPathAlgorithm
Apr4,2016 CSCI211-Sprenkle 15
Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow Gf = residual graph
while there exists augmenting path P f = Augment(f, c, P) # change the flow update Gf # build a new residual graph
return f
Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(eR) = f(e) - b # forward edge, ê flow return f
Why does alg work? What is happening at each iteration?What is the running time? Need more analysis …
MINIMUMCUTS
Apr4,2016 CSCI211-Sprenkle 16
Cuts• Ans-tcutisapar&&on(A,B)ofVwiths∈Aandt∈B
• Thecapacityofacut(A,B)is
Apr4,2016 CSCI211-Sprenkle 17
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€
cap( A, B) = c(e)e out of A∑
Capacity = 9 + 15 + 8 + 30� = 62
BWhat is the capacity of this cut?
MinimumCutProblem• Findans-tcutofminimumcapacity
Ø Putsupperboundonmaximumflow
Apr4,2016 CSCI211-Sprenkle 18
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A Capacity = 10 + 8 + 10� = 28
B
Same graph, different cut
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Recall• Thevalueofaflowfisv(f)=∑eoutofsf(e)
Apr4,2016 CSCI211-Sprenkle 19
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Value = 40
capacityflow
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FlowValueLemma• Letfbeanyflow,andlet(A,B)beanys-tcut.Then,thevalueoftheflowis=fout(A)–fin(A).
Apr4,2016 CSCI211-Sprenkle 20
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Value = 24
€
f (e)e out of A∑ − f (e)
e in to A∑ = v( f )
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B
What is the value of this flow?
FlowValueLemma
Apr4,2016 CSCI211-Sprenkle 21
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Value = 6 + 0 + 8 - 1 + 11� = 24
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• Letfbeanyflow,andlet(A,B)beanys-tcut.Then,thevalueoftheflowis=fout(A)–fin(A).
€
f (e)e out of A∑ − f (e)
e in to A∑ = v( f )
CSCI211-Sprenkle
Possibilities for edge e:• Both ends in A (0)• Points out from A (+), Points in to A (-)
FlowValueLemma(FVL)• Letfbeanyflow,andlet(A,B)beanys-tcut.• Then• Pf.
Apr4,201622
by flow conservation, �all terms except v = s are 0
By definition
A B
WeakDuality• Letfbeanyflowandlet(A,B)beanys-tcut.➜ Thenthevalueoftheflowisatmostthecut’scapacity
Apr4,2016 CSCI211-Sprenkle 23
Cut capacity = 30 ⇒ Flow value ≤ 30
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Capacity = 30
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WeakDuality• Letfbeanyflow.Then,foranys-tcut(A,B)v(f)≤cap(A,B).
• Pf.
Apr4,2016 CSCI211-Sprenkle 24
€
v( f ) = f (e)e out of A∑ − f (e)
e in to A∑
≤ f (e)e out of A∑
≤ c(e)e out of A∑
= cap(A,B) s
t
A B
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84By FVL
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Cer&ficateofOp&mality• Corollary.Letfbeanyflow,andlet(A,B)beanycut.Ifv(f)=cap(A,B),thenfisamaxflowand(A,B)isamincut.
Apr4,2016 CSCI211-Sprenkle 25
Value of flow = 28�Cut capacity = 28 ⇒�
Flow value ≤ 2810
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Recall:ResidualGraphGf• Originaledge:e=(u,v)∈E
Ø Flowf(e),capacityc(e)• Residualedge
Ø e=(u,v)w/capacityc(e)-f(e)Ø eR=(v,u)withcapacityf(e)
• Toundoflow• Residualgraph:Gf=(V,Ef)
Ø Residualedgeswithposi(veresidualcapacityØ Ef={e:f(e)<c(e)}∪{eR:f(e)>0}
Apr4,2016 CSCI211-Sprenkle 26
u v 11
residual capacity
6
residual capacity
Forward edges Backward edges
u v 17
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capacity
flow
Recall:Augmen&ngPathAlgorithm
Apr4,2016 CSCI211-Sprenkle 27
Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow Gf = residual graph
while there exists augmenting path P f = Augment(f, c, P) # change the flow update Gf # build a new residual graph
return f
Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(eR) = f(e) - b # forward edge, ê flow return f
Intui&onBehindCorrectnessofF-FAlgorithm
• LetAbesetofver&cesreachablefromsinresidualgraphatendofF-Falgexecu&on
• Bydefini&onofA,s∈A• Bydefini&onoftheF-Falgorithm’sresul&ngflow,t∉A
Apr4,2016 CSCI211-Sprenkle 28
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 29
s
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Flow value = 19Cut capacity = 19
A
• What do we know about the flow out of A?• What do we know about the flow into A?
A: nodes reachable from s
A
Ford-FulkersonAlgorithm
Apr4,2016 CSCI211-Sprenkle 30
s
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Flow value = 19Cut capacity = 19
• What do we know about the flow out of A?• What do we know about the flow into A?
A
• All edges out of A are completely saturated• All edges into A are completely unused
A
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Max-FlowMin-CutTheorem
• Proofstrategy.Provebothsimultaneouslybyshowingthefollowingareequivalent:(i)Thereexistsacut(A,B)suchthatv(f)=cap(A,B).(ii)Flowfisamaxflow.(iii)Thereisnoaugmen&ngpathrela&vetof. 31Apr4,2016 CSCI211-Sprenkle
Max-flow min-cut theorem. [Ford-Fulkerson 1956] �The value of the max flow is equal to the value of the min cut.
Augmenting path theorem. �Flow f is a max flow iff there are no augmenting paths.
See formal proof in book
AnalyzingAugmen&ngPathAlgorithm
Apr4,2016 CSCI211-Sprenkle 32
Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow Gf = residual graph
while there exists augmenting path P f = Augment(f, c, P) # change the flow update Gf # build a new residual graph
return f
Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(eR) = f(e) - b # forward edge, ê flow return f
Costs?
AnalyzingAugmen&ngPathAlgorithm
Apr4,2016 CSCI211-Sprenkle 33
Ford-Fulkerson(G, s, t, c) foreach e ∈ E f(e) = 0 # initially no flow Gf = residual graph
while there exists augmenting path P f = Augment(f, c, P) # change the flow update Gf # build a new residual graph
return f
Augment(f, c, P) b = bottleneck(P) # edge on P with least capacity foreach e ∈ P if (e ∈ E) f(e) = f(e) + b # forward edge, é flow else f(eR) = f(e) - b # forward edge, ê flow return f
O(m)
O(m)
O(m)
O(m)
O(n)O(n)
O(1)O(1)
Total:O(n)àO(m),sincen≤2m
Total:O(Fm)
Find path: O(m); Iterations: O(F) iterations, where F = max flow
RunningTime• Assump&on.Allcapaci&esareintegersbetween1andF.• Invariant.Everyflowvaluef(e)andeveryresidualcapacity’s
cf(e)remainsanintegerthroughoutalgorithm.
• Theorem.Algorithmterminatesinatmostv(f*)≤nFitera&ons.• Pf.Eachaugmenta&onincreasesvaluebyatleast1.• Corollary.IfF=1,Ford-FulkersonrunsinO(mn)&me.
• Integralitytheorem.Ifallcapaci&esareintegers,thenthereexistsamaxflowfforwhicheveryflowvaluef(e)isaninteger.
• Pf.Sincealgorithmterminates,theoremfollowsfrominvariant.
34Apr4,2016 CSCI211-Sprenkle
LookingAhead
• Wiki:Duetonight(7.1-7.2,7.5,7.7)Ø 7.5won’tbediscussedinclass
• ProblemSet9dueFriday
Apr4,2016 CSCI211-Sprenkle 35
