[SOLVED] CS data mining ER algorithm Lecture 8

Lecture 8
Flow networks I
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General techniques in this course
Greedy algorithms [Lecture 3]
Divide & Conquer algorithms [Lectures 4 and 5]
Dynamic programming algorithms [Lectures 6 and 7] Network flow algorithms [today and 2 May]
Theory [today]
Applications [2 May]
NP and NP-completeness Coping with hardness
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Soviet Rail Network, 1955
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Reference: On the history of the transportation and maximum flow problems. Alexander Schrijver in Math Programming, 91: 3, 2002.

Maximum Flow and Minimum Cut
Max flow and min cut.
Two very rich algorithmic problems.
Cornerstone problems in combinatorial optimization.
Mathematical duality.
Nontrivial applications / reductions.
Data mining.
Open-pit mining.
Project selection.
Airline scheduling.
Bipartite matching.
Baseball elimination.
Image segmentation.
Network connectivity.
Network reliability.
Distributed computing.
Egalitarian stable matching.
Securityofstatisticaldata.
Network intrusion detection.
Multi-camera scene reconstruction.
Manymanymore
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Flow network
Abstraction for material flowing through the edges.
G = (V, E): a directed graph with no parallel edges.
Two distinguished nodes: s = source, t = sink.
The source has no incoming edges and the sink has no outgoing edges.
c(e) = capacity of edge e. 295
10
sources 5 3 8 6
4
15 15
10
10 tsink 10
Page 6
capacity
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15
46
15 4 30 7

Flows
Definition: An s-t flow is a function that satisfies: For each e I E: 0 f(e) c(e)
We say e is saturated if f(e) = c(e)
For each v I V {s, t}: a f(e) =
(capacity) (conservation)
e in to v
Definition: The value of a flow f is:
a f(e) e out of v
v( f ) = 2 9 5
a f (e) . e out of s
0
10
4
0
10
capacity 10 flow 4
0
15 15 0
44
s 5 3 8 6 10 t
04
15
0
40 6 150 0
4 30 7
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0
Value =P4age 7
0

Flows
Definition: An s-t flow is a function that satisfies: For each e I E: 0 f(e) c(e)
We say e is saturated if f(e) = c(e)
For each v I V {s, t}: a f(e) =
(capacity) (conservation)
e in to v
Definition: The value of a flow f is:
a f(e) e out of v
v( f ) = 2 9 5
a f (e) . e out of s
6
10
8
6
capacity 10
15 15 0 0
44
s 5 3 8 6 10 t
flow 10 38
15
11
40 6 150 1
4 30 7
10
10
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11
Value = 2Pa4ge 8

Maximum Flow Problem
Max flow problem. Find s-t flow of maximum value.
Question: How to characterize optimal solution?
DP and D&C uses a recurrence equation. Not known if max flows admit such a recurrence.
2 9 5
9
capacity 10
15 15 0 1
9
10
9
40
s 5 3 8 6 10 t
flow 10 48
15
14
40 6 150 4
4 30 7
10
10
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14
Value = 2Pa8ge 9

Characterizing Max Flow
f is a max flow if:
f saturates every edge OR
f saturates every edge out of s
2 9 5
9
capacity 10
15 15 0 1
9
10
9
40
s 5 3 8 6 10 t
flow 10 48
15
14
40 6 150 4
4 30 7
10
10
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14
Value =P2ag8e 10

Cuts
Definitions:
Ans-tcutisapartition(A,B)ofVwithsIAandtIB.
The capacity of a cut (A, B) is: cap(A, B) = 295
a c(e) e out of A
10
4
15 15
10
s 5 3 8 6 10 t A
15
46
10
15 4 30 7
Capacity = 10 + 5 + 15
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= 30
Page 11

Cuts
Definitions:
Ans-tcutisapartition(A,B)ofVwithsIAandtIB.
The capacity of a cut (A, B) is: cap(A, B) = 2 9 5
10
a c(e) e out of A
15 15
s 5 3 8 6 10 t
4
10
A
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15 4 30 7
15
46
10
Capacity = 9 + 15 + 8 + 30
= 62
Page 12

Minimum Cut Problem
Min s-t cut problem:
Find an s-t cut of minimum capacity.
Question: How to characterize optimal solution?
Not known if min cuts admit a DP-style recurrence.
2 9 5
10
4
15
s 5 3 8 6 10 t
15
10
A
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15
46
4 30 7
10
15
Capacity = 10 + 8 + 10
= 28
Page 13

Max-Flow = Min-Cut
1. Max-Flow Min-Cut
2. AlgorithmforMax-Flowfindsaflowfandacut(A,B)such that v(f) = cap(A,B)
2 9 5
9
capacity 10
15 15 0 1
9
10
9
40
s 5 3 8 6 10 t
flow 10 48
15
14
40 6 150 4
4 30 7
10
10
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14
Value =P2ag8e 14

Notation
Given a vertex u, define fout(u) = total flow on edges leaving u
Given a vertex subset S, define fout(S) = total flow on edges leaving S
Similarly, define fin(u) and fin(S) on edges entering u and S, resp.
Can rewrite v(f) = fout(s) and fin(u) = fout(u) for v not s,t
6
2 9 5
10
10
44
0
15 15 0
6
10
388
s 5 3 8 6 10 t
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4 30 7
A
1
40 6 150
11
10
10
15
11
Value = 10+3+11 = 24Page 15

Flows and Cuts
Flow value lemma. Let f be any flow, and let (A, B) be any s-t cut. Then, the net flow sent across the cut is equal to the amount
leaving s.
v(f) = fout(A) fin(A) 6
2 9 5
10
10
44
0
15 15 0
6
10
388
s 5 3 8 6 10 t
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4 30 7
A
1
40 6 150
11
10
10
15
11
Value = 10+3+11 = 24Page 18

Flows and Cuts
Flow value lemma. Let f be any flow, and let (A, B) be any s-t cut. Then, the net flow sent across the cut is equal to the amount
leaving s.
v(f) = fout(A) fin(A) 6
2 9 5
10
10
44
0
15 15 0
6
10
388
s 5 3 8 6 10 t
A
1
40 6 150
11
10
10
15
11
Value = 6 + 0 + 8 1 + 11 = 24 Page 19
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4 30 7

Flows and Cuts
Flow value lemma. Let f be any flow, and let (A, B) be any s-t cut. Then, the net flow sent across the cut is equal to the amount
leaving s.
v(f) = fout(A) fin(A) 6
2 9 5
10
10
44
0
15 15 0
6
10
388
s 5 3 8 6 10 t
A
1
40 6 150
11
10
10
15
11
Value = 10 4 + 8 0 + 10 = 24 Page 20
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4 30 7

Flowvaluelemmacziven flow f
f
f 174
finfu FOH
f sforetCA Pt by induction on IAI
Inductive case A I Let u c Aks
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Equiv
WTS
f
want
f UtfAl u3 fin fat A fin A
fat CA fotCAl a3
Alfie
finCA finCA
cut A B fatCA fin A
fztfz
re
f fout EB finS3
s
to
A 1243
O
Inc in fin 4 11
Increase in fo
tz f Flow conservation
fg fg

Flows and Cuts
Weak duality. Let f be any flow, and let (A, B) be any s-t cut. Then the value of the flow is at most the capacity of the cut, i.e.
v(f) cap(A, B)
Cut capacity = 30 Flow value 30 2 9 5
10
4
15 15
10
s 5 3 8 6 10 t A
15
46
10
15 4 30 7
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Capacity = P3ag0e 23

Flows and Cuts
Weak duality. Let f be any flow, and let (A, B) be any s-t cut. Then the value of the flow is at most the capacity of the cut, i.e.
v(f) cap(A, B)
Cut capacity = 28 Flow value 28 2 9 5
10
4
15 15
10
s 5 3 8 6 10 t A
15 4 30 7
15
46
10
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Capacity = P2ag8e 24

Flows and Cuts
Weak duality. Let f be any flow, and let (A, B) be any s-t cut. Then the value of the flow is at most the capacity of the cut, i.e.,
Proof:
v(f) = =
=
fout(A) fin(A)
fout(A)
S f(e) e out
of A
Sc(e) e out
of A c(A,B)
A
8
B
t
v(f) cap(A, B).
4
s
7
6
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Certificate of Optimality
Corollary: Let f be any flow, and let (A, B) be any cut.
If v(f) = cap(A, B) then f is a max flow and (A, B) is a min cut.
Value of flow = 28
Cut capacity = 28 Flow value 28
9
2 9 5
10
10
1
15 15 0
9
10
40 489
s 5 3 8 6 10 t
4
A 15 40 6 150
10
10
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4 30 7
Page 26
14
14

Summary (so far)
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1. Max flow problem
2. Min cut problem
3. Theorem: Max flow Min cut

Towards a Max Flow Algorithm
Greedy algorithm.
Startwithf(e)=0foralledgeeIE.
Find an s-t path P where each edge has f(e) < c(e). Augment flow along path P. Repeat until you get stuck. 10020 10s 300 t10 2000Flow value = 0 The University of SydneyPage 282 Towards a Max Flow AlgorithmGreedy algorithm. Startwithf(e)=0foralledgeeIE. Find an s-t path P where each edge has f(e) < c(e). Augment as much flow as possible along path P. Repeat until you get stuck. 1 s s3 0 X0 2 0t t20 X02001010020X0 2 0Flow value = 20 The University of SydneyPage 292 Towards a Max Flow AlgorithmAugmenting greedy flow to get optimal flow1120 0 20 1020 10 20 10s 3020 t s 3010 t10 20 10 200 20 10 2022TgherUeniverdsityof=Syd2ne0y opt = 30 Page 30 Towards a Max Flow AlgorithmAugmenting greedy flow to get optimal flow Send 10 units on (s,2) edge1120 0 20 1020 10 20 10s 3020 t s 3010 t10 20 10 2010 20 10 2022TgherUeniverdsityof=Syd2ne0y opt = 30 Page 31 Towards a Max Flow AlgorithmAugmenting greedy flow to get optimal flow Send 10 units on (s,2) edge Undo 10 units on (1,2) edge to preserve conservation at vertex 21120 0 20 1020 10 20 10s 3010 t s 3010 t10 20 10 2010 20 10 2022TgherUeniverdsityof=Syd2ne0y opt = 30 Page 32 Towards a Max Flow AlgorithmAugmenting greedy flow to get optimal flow Send 10 units on (s,2) edge Undo 10 units on (1,2) edge to preserve conservation at vertex 2 Send 10 units on (1,t) edge to preserve conservation at vertex 11120 10 20 1020 10 20 10s 3010 t s 3010 t10 20 10 2010 20 10 2022TgherUeniverdsityof=Syd2ne0y opt = 30 Page 33 Build a Residual Graph Gf = (V, Ef ) Original edge: e = (u, v) I E. Flow f(e), capacity c(e). Residual edge. “Undo” flow sent. e=(u,v)andeR =(v,u). Residual capacity:cf(e)=iic(e)-f(e) ifeIE if(e) if eR IEcapacityu 17 v6flowresidual capacity vresidual capacity u 11 6 Residual graph: Gf = (V, Ef ). Residual edges with positive residual capacity. Ef ={e:f(e) 0}.
Max flow of Gf = (Max flow of G) v(f) (Exercise) The University of Sydney
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Build a Residual Graph Gf = (V, Ef )
The residual capacity of an edge in Gf tells us how much flow we can send, given the current flow.
residual capacity
u 11 v
6
residual capacity
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Augmenting Path Algorithm
Notations:
P = a simple s-t path in Gf
bottleneck(P,f) = minimum residual capacity of any edge on P with respect to the current flow f.
u 20
G
s
10
10 30
t 20
v
u 10 Gf 20
s 30 10
v
t 20
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Augmenting Path Algorithm
Notations:
P = a simple s-t path in Gf
bottleneck(P,f) = minimum residual capacity of any edge on P with respect to the current flow f.
0/10
s
v
u 20/30
G
20/20 s
0/10
t 20/20
v
u Gf
t
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Page 37

Augmenting Path Algorithm
Notations:
P = a simple s-t path in Gf
bottleneck(P,f) = minimum residual capacity of any edge on P with respect to the current flow f.
u 20/30
20/20 s
0/10
v
0/10
G
t 20/20
u Gf 10
20
s 10
v
20 10
t 20
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Page 38

Augmenting Path Algorithm
Notations:
P = a simple s-t path in Gf
bottleneck(P,f) = minimum residual capacity of any edge on P with respect to the current flow f.
u 20/30
G
t 20/20
u Gf 10
Augment(f,P) {
b bottleneck(P,f) foreach e =(u,v) I P {
if e is a forward edge then
increase f(e) in G by b
else (e is a backward edge)
decrease f(e) in G by b
}
return f }
20/20 s
0/10
v
0/10
20
s 10
v bottleneck
20 10
t 20
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Page 39

Augmenting Path Algorithm
Notations:
P = a simple s-t path in Gf
bottleneck(P,f) = minimum residual capacity of any edge on P with respect to the current flow f.
u s 10/30
G
Augment(f,P) {
b bottleneck(P,f) foreach e =(u,v) I P {
if e is a forward edge then
increase f(e) in G by b
else (e is a backward edge)
decrease f(e) in G by b
}
return f }
20/20
t
10/10
10/10
v
u Gf 10
20/20
20
s 10
v
20 10
t 20
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Augment(f,P) gives a new flow f in G

Augmenting Path Algorithm
Ford-Fulkerson(G,s,t) { foreach e I E
f(e) 0
Gf residual graph
while (there exists augmenting path P in Gf){ f Augment(f,P)
update Gf }
return f }
Augment(f,P) {
b bottleneck(P,f) foreach e =(u,v) I P {
if e is a forward edge then
increase f(e) in G by b
else (e is a backward edge)
decrease f(e) in G by b
}
return f }
Page 41
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Ford-Fulkerson Algorithm
244
10 2 8 6 10
s 10 3 9 5 10 t
G:
capacity
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Ford-Fulkerson Algorithm
0
2 4 4
flow
capacity
G:
s 10 3 9 5 10 t
0 10208 6010
00 000
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Flow value = 0

Ford-Fulkerson Algorithm
0
2 4 4
flow
capacity
G:
s 10 3 9 5 10 t
0 10208 6010
8X0 0 X8
0 0 8X0
Flow value = 0
244
residual capacity 10
Gf:
10
2
86
s 10 3 9 5 10 t
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Ford-Fulkerson Algorithm
G:
0
2 4 4
1 0 X8 8
10208 6010
s 10 3 X 5 t 9 10
X
0 2 02 10X8
0
2 4 4
Flow value = 8
Gf:
8
2
86
10
2 s 10 3
9 5 2 t 8
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Ford-Fulkerson Algorithm
G:
0
2 4 4
X0 6 1022 X10
X06
s 10 3 X 5
2 4 4
10 2 8 6 10
10 8
860
28
9 10
t
Flow value = 10
6 10
Gf:
s 10 3 7 5 10 t 2
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Ford-Fulkerson Algorithm
2 4 4
02 X
G:
s 10 3 9 5 10 t
X6 8 10 22 8 66 10
10 8 X0
X68 8 10
Flow value = 16
Gf:
2 4 4
10 2 8 6 4
6
s 4 3 1 5 10 t
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6
8

Ford-Fulkerson Algorithm
G:
23 X
2 4 4
X8 9 10 20 8 66 10
10 87 X
X8 9 8 9 1 0 s 10 3 X 5
t
Flow value = 18
8
9 10
Gf:
2
2 2 4
10 2 8 6 2
s 2 3 1 5 10 t
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8
8

Ford-Fulkerson Algorithm
3
2 4 4
G:
s 10 3 9 5 10 t
9 10 20 8 66 10
10 7
9 9 10
3
2 1 4
Gf: 1 9
Flow value = 19
10 2 7
6 1
s 1 3 9 5 10 t 9
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Page 49

Ford-Fulkerson Algorithm
3
2 4 4
G:
s 10 3 9 5 10 t
9 10 20 8 66 10
10 7
9 9 10
Cut capacity = 19 Flow value = 19
3
2 1 4
Gf: 1 9
10 2 7
6 1
s 1 3 9 5 10 t 9
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Page 50

Augmenting Path Algorithm
Ford-Fulkerson(G,s,t) { foreach e I E
f(e) 0
Gf residual graph
while (there exists augmenting path P in Gf){ f Augment(f,P)
update Gf }
return f }
Augment(f,P) {
b bottleneck(P,f) foreach e =(u,v) I P {
if e is a forward edge then
increase f(e) in G by b
else (e is a backward edge)
decrease f(e) in G by b
}
return f }
Page 51
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Max-Flow Min-Cut Theorem
Augmenting path theorem: Flow f is a max flow if and only if there are no augmenting paths.
Max-flow min-cut theorem: The value of the max flow is equal to the value of the min cut. [Ford-Fulkerson 1956]
Proof strategy: We prove both simultaneously. Let f be a flow. Then the following are equivalent:
(i) There exists a cut (A, B) such that v(f) = cap(A, B). (ii) Flow f is a max flow.
(iii) There is no augmenting path relative to f.
(i) (ii) This was the corollary to the weak duality lemma.
(ii) (iii) We show contrapositive.
Letfbeaflow.Ifthereexistsanaugmentingpath,thenwecanimprovef
by sending flow along a path P and augment the flow in G.
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Proof of Max-Flow Min-Cut Theorem
(iii) (i)
Let f be a flow with no augmenting paths.
Let A be set of vertices reachable from s in residual graph. BydefinitionofA,sIA.
Bydefinitionoff,tIA.
v(f)= X f(e) X f(e) e out of A e into A
A
B
t
s
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original network
Page 53

Proof of Max-Flow Min-Cut Theorem
(iii) (i)
Let f be a flow with no augmenting paths.
Let A be set of vertices reachable from s in residual graph. BydefinitionofA,sIA.
Bydefinitionoff,tIA.
v(f)= X f(e) X f(e) e out of A e into A
A
B
t
No augmenting path from A to B => every edge leaving A saturated, every edge entering A is empty
s
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original network
Page 54

Proof of Max-Flow Min-Cut Theorem
(iii) (i)
Let f be a flow with no augmenting paths.
Let A be set of vertices reachable from s in residual graph. BydefinitionofA,sIA.
Bydefinitionoff,tIA.
v(f)= X f(e) X f(e) e out of A e into A
= c(e) e out of a
= cap(A, B)
No augmenting path from A to B => every edge leaving A saturated, every edge entering A is empty
A
B
t
s
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original network
Page 55

Max-Flow Min-Cut Theorem
Augmenting path theorem: Flow f is a max flow if and only if there are no augmenting paths.
Max-flow min-cut theorem: The value of the max flow is equal to the value of the min cut. [Ford-Fulkerson 1956]
Proof strategy: We prove both simultaneously. Let f be a flow. Then the following are equivalent:
(i) There exists a cut (A, B) such that v(f) = cap(A, B). (ii) Flow f is a max flow.
(iii) There is no augmenting path relative to f.
Note: This implies we can check if a given flow f is max flow in time O(n + m)!
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Ford-Fulkerson: Analysis
Assumption. All initial capacities are integers.
Lemma. At every intermediate stage of the Ford-Fulkerson algorithm
the flow values and the residual graph capacities in Gf are integers.
Proof: (proof by induction)
Base case: Initially the statement is correct. Induction hyp.: True after j iterations.
Induction step: Since all the residual capacities in Gf are integers the bottleneck-value must be an integer. Thus the flow will have integer values and hence also the capacities in the new residual graph.
Integrality theorem. If all capacities are integers, then there exists a max flow f for which every flow value f(e) is an integer.
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Ford-Fulkerson: Running Time
Observation:
Let f be a flow in G, and let P be a simple s-t path in Gf. v(f) = v(f) + bottleneck(f,P)
and since bottleneck(f,P)>0 v(f) > v(f).
The flow value strictly increases in an augmentation
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Ford-Fulkerson: Running Time
Notation: C = S c(e) e out
of s
Observation: C is an upper bound on the maximum flow.
Theorem. The algorithm terminates in at most v(fmax) C iterations. Proof: Each augmentation increase flow value by at least 1.
5
2
8
s
3
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Ford-Fulkerson: Running Time
Corollary:
Ford-Fulkerson runs in O((m+n)C) time, if all capacities are integers.
Proof: C iterations.
Path in Gf can be found in O(m+n) time using BFS. Augment(P,f) takes O(n) time.
Updating Gf takes O(n) time.
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7.3 Choosing Good Augmenting Paths
Is O(C(m+n)) a good time bound?
Yes, if C is small.
If C is large, can the number of iterations be as bad as C?
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Ford-Fulkerson: Exponential Number of Augmentations
Question: Is generic Ford-Fulkerson algorithm polynomial in
input size?
Answer: No. If max capacity is D, then algorithm can take D iterations.
1
1X0 0
DD
s 1X01 t
m, n, and log C
D
D
00 X1
2
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Ford-Fulkerson: Exponential Number of Augmentations
Question: Is generic Ford-Fulkerson algorithm polynomial in
input size?
Answer: No. If max capacity is D, then algorithm can take D iterations.
11
1 X0 0 1 X0 X0 1 DDDD
s 1X01 t s 1X0X10 t DDDD
0 0 10 X01 X1X
m, n, and log C
22
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Choosing Good Augmenting Paths
Use care when selecting augmenting paths.
Some choices lead to exponential algorithms.
Clever choices lead to polynomial algorithms.
If capacities are irrational, algorithm not guaranteed to terminate!
Goal: choose augmenting paths so that: Can find augmenting paths efficiently.
Few iterations.
Choose augmenting paths with: [Edmonds-Karp 1972, Dinitz 1970] Max bottleneck capacity.
Sufficiently large bottleneck capacity.
Fewest number of edges.
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Choosing Good Augmenting Paths
Ford Fulkerson
Choose any augmenting path (C iterations)
Edmonds Karp #1 (m log C iterations) Choose max flow path
Improved Ford Fulkerson via capacity scaling (log C iterations) Choose max flow path
Edmonds Karp #2 (O(nm) iterations)
Choose minimum link path [Edmonds-Karp 1972, Dinitz 1970]
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Edmonds-Karp #1
Pick the augmenting path with largest capacity [maximum bottleneck path]
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Edmonds-Karp #1
Pick the augmenting path with largest capacity [maximum bottleneck path]
Claim: If maximum flow in G is F, there must exists a path from s to t with bottleneck capacity at least F/m.
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Edmonds-Karp #1
Pick the augmenting path with largest capacity [maximum bottleneck path]
Claim: If maximum flow in G is F, there must exists a path from s to t with bottleneck capacity at least F/m.
Proof:
Delete all edges of capacity less than F/m.
Is the graph still connected?
F=24 m=15
6 10 t
2 9
15 s 5 3 8
5
15 10
10
4
15
4 6 15 4 30 7
10
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Page 68

Edmonds-Karp #1
Pick the augmenting path with largest capacity [maximum bottleneck path]
Claim: If maximum flow in G is F, there must exists a path from s to t with bottleneck capacity at least F/m.
Proof:
Delete all edges of capacity less than F/m.
Is the graph still connected?
Yes, otherwise we have a cut of value less than F.
A
B
t
< F/m s The University of Sydney< F/mPage 69 Edmonds-Karp #1Pick the augmenting path with largest capacity [maximum bottleneck path]Claim: If maximum flow in residual graph Gf is F, there must exists a path from s to t with bottleneck capacity at least F/m.Proof:Delete all edges of capacity less than F/m.Is the graph still connected?Yes, otherwise we have a cut of value less than F.ABt< F/m s The University of Sydney< F/mPage 70 Edmonds-Karp #1Theorem: Edmonds-Karp #1 makes at most O(m log F) iterations.Proof:At least 1/m of remaining flow is added in each iteration.URemaining flow reduced by a factor of (1-1/m) per iteration. #iterations until remaining flow <1? F(1-1/m)x <1?We know: (1-1/m)m < 1/eSet x = m ln F F (1-1/m)m ln F < F (1/e)ln F = 1 The University of SydneyPage 73 ApplicationsThe University of SydneyPage 74 Bipartite matching Perfect matching Disjoint paths Network connectivity Circulation problems Image segmentation Baseball elimination Project selection SummaryThe University of SydneyPage 751. 2. 3.4. 5.Max flow problemMin cut problemFord-Fulkerson:1. Residual graph 2. correctness3. complexityMax-Flow Min-Cut theorem Edmonds-Karp Appendix: Alternate proof of flow value lemmaFlow value lemma. Let f be any flow, and let (A, B) be any s-t cut. Then, the net flow sent across the cut is equal to the amountleaving s.Proof:v(f) = fout(A) fin(A)v(f) = fout(s) = fout(s) fin(s)by flow conservation, all terms except v = s are 0, i.e. fout(v) – fin(v) = 0= S (fout(v) fin(v)) vIA=S f(e)Sf(e) The University of SydneyPage 76e out of Ae into A = fout(A) fin(A)