taken from kevin wayne’s slides (princeton university) cosc 3101a - design and analysis of...

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Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

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3 n Network reliability. n Security of statistical data. n Distributed computing. n Egalitarian stable matching. n Distributed computing. n Many many more... Maximum Flow and Minimum Cut Max flow and min cut. n Two very rich algorithmic problems. n Cornerstone problem in combinatorial optimization. n Beautiful mathematical duality. Nontrivial applications / reductions. n Network connectivity. n Bipartite matching. n Data mining. n Open-pit mining. n Airline scheduling. n Image processing. n Project selection. n Baseball elimination.

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Page 1: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

Taken from Kevin Wayne’s slides (Princeton University)

COSC 3101A - Design and Analysis of Algorithms

13

Maximum Flow

Page 2: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

2

Contents

Contents. Maximum flow problem. Minimum cut problem. Max-flow min-cut theorem. Augmenting path algorithm. Capacity-scaling. Shortest augmenting path.

Page 3: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

3

Network reliability. Security of statistical data. Distributed computing. Egalitarian stable matching. Distributed computing. Many many more . . .

Maximum Flow and Minimum Cut

Max flow and min cut. Two very rich algorithmic problems. Cornerstone problem in combinatorial optimization. Beautiful mathematical duality.

Nontrivial applications / reductions. Network connectivity. Bipartite matching. Data mining. Open-pit mining. Airline scheduling. Image processing. Project selection. Baseball elimination.

Page 4: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

4

Max flow network: G = (V, E, s, t, u) . (V, E) = directed graph, no parallel arcs. Two distinguished nodes: s = source, t = sink. u(e) = capacity of arc e.

Max Flow Network

Capacity

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

Page 5: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

5

An s-t flow is a function f: E that satisfies: For each e E: 0 f(e) u(e) (capacity) For each v V – {s, t}: (conservation)

Flows

veveefef

ofout to in )()(

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

4

0

0

0

0

0 0

0 4 4

04 0

00Capacity

Flow

Ewvwve

wvfef),(: ofout

),(:)(Evwwve

vwfef),(: to in

),(:)(

Page 6: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

6

An s-t flow is a function f: E that satisfies: For each e E: 0 f(e) u(e) (capacity) For each v V – {s, t}: (conservation)

MAX FLOW: find s-t flow that maximizes net flow out of the source.

Flows

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

4

0

0

0

0

0 0

0 4 4

04 0

Value = 4

seeff

ofout )(

00Capacity

Flow

veveefef

ofout to in )()(

Page 7: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

7

An s-t flow is a function f: E that satisfies: For each e E: 0 f(e) u(e) (capacity) For each v V – {s, t}: (conservation)

MAX FLOW: find s-t flow that maximizes net flow out of the source.

Flows

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

10

6

6

11

11

1 10

3 8 8

04 0

Value = 2400

Capacity

Flow

veveefef

ofout to in )()(

Page 8: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

8

An s-t flow is a function f: E that satisfies: For each e E: 0 f(e) u(e) (capacity) For each v V – {s, t}: (conservation)

MAX FLOW: find s-t flow that maximizes net flow out of the source.

Flows

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

10

9

9

14

14

4 10

4 8 9

10 0

Value = 2800

Capacity

Flow

veveefef

ofout to in )()(

Page 9: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

9

Networks

communication

Network

telephone exchanges,computers, satellites

Nodes Arcs

cables, fiber optics,microwave relays

Flow

voice, video,packets

circuits gates, registers,processors wires current

mechanical joints rods, beams, springs heat, energy

hydraulic reservoirs, pumpingstations, lakes pipelines fluid, oil

financial stocks, currency transactions money

transportation airports, rail yards,street intersections

highways, railbeds,airway routes

freight,vehicles,passengers

chemical sites bonds energy

Page 10: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

10

An s-t cut is a node partition (S, T) such that s S, t T. The capacity of an s-t cut (S, T) is:

Min s-t cut: find an s-t cut of minimum capacity.

Cuts

TwSvEwvSe

wvueu

,),( ofout

).,(:)(

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

Capacity = 30

Page 11: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

11

An s-t cut is a node partition (S, T) such that s S, t T. The capacity of an s-t cut (S, T) is:

Min s-t cut: find an s-t cut of minimum capacity.

Cuts

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

Capacity = 62

TwSvEwvSe

wvueu

,),( ofout

).,(:)(

Page 12: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

12

An s-t cut is a node partition (S, T) such that s S, t T. The capacity of an s-t cut (S, T) is:

Min s-t cut: find an s-t cut of minimum capacity.

Cuts

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

Capacity = 28

TwSvEwvSe

wvueu

,),( ofout

).,(:)(

Page 13: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

13

L1. Let f be a flow, and let (S, T) be a cut. Then, the net flow sent across the cut is equal to the amount reaching t.

Flows and Cuts

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

fefefefseSeSe

ofout to in ofout

)()()(

Value = 24

10

6

6

10

10

0 10

4 8 8

04 0

00

Page 14: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

14

10

6

6

10

10

0 10

4 8 8

04 0

00

L1. Let f be a flow, and let (S, T) be a cut. Then, the net flow sent across the cut is equal to the amount reaching t.

Flows and Cuts

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

fefefefseSeSe

ofout to in ofout

)()()(

Value = 24

Page 15: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

15

10

6

6

10

10

0 10

4 8 8

04 0

0

L1. Let f be a flow, and let (S, T) be a cut. Then, the net flow sent across the cut is equal to the amount reaching t.

Flows and Cuts

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

fefefefseSeSe

ofout to in ofout

)()()(

Value = 24

0

Page 16: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

16

Let f be a flow, and let (S, T) be a cut. Then,

Proof by induction on |S|. Base case: S = { s }. Inductive hypothesis: assume true for |S| < k.

– consider cut (S, T) with |S| = k– S = S' { v } for some v s, t, |S' | = k-1 cap(S', T') = | f |.– adding v to S' increase cut capacity by

Flows and Cuts.)()(

to in ofout

Se Sefefef

0)()( to in ofout

veveefef

s

vt

After

s

vt

S' Before S

Page 17: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

17

L2. Let f be a flow, and let (S, T) be a cut. Then, | f | cap(S, T).

Proof.

Corollary. Let f be a flow, and let (S, T) be a cut. If |f| = cap(S, T), then f is a max flow and (S, T) is a min cut.

Flows and Cuts

T)cap(S,

)(

)(

)()(

ofout

ofout

to in ofout

Se

Se

SeSe

eu

ef

efeff

s

t

S T

76

84

Page 18: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

18

Max Flow and Min Cut

Corollary. Let f be a flow, and let (S, T) be a cut. If |f| = cap(S, T), then f is a max flow and (S, T) is a min cut.

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

10

9

9

15

15

4 10

4 8 9

10 0

Flow value = 28

00

Cut capacity = 28

Page 19: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

19

Max-Flow Min-Cut Theorem

MAX-FLOW MIN-CUT THEOREM (Ford-Fulkerson, 1956): In any network, the value of the max flow is equal to the value of the min cut.

"Good characterization." Proof IOU.

s

2

3

4

5

6

7

t

15

5

30

15

10

8

15

9

6 10

10

10 15 4

4

10

9

9

15

15

4 10

4 8 9

10 0

Flow value = 28

00

Cut capacity = 28

Page 20: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

20

Towards an Algorithm

Find an s-t path where each arc has u(e) > f(e) and "augment" flow along the path.

s

4

2

5

3 t

4

0 0

0 0 0

0

40

4 4

Flow value = 0

10 13 10

Page 21: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

21

Towards an Algorithm

Find an s-t path where each arc has u(e) > f(e) and "augment" flow along the path.

Repeat until you get stuck.

s

4

2

5

3 t

4

0 0

0 0 0

0

40

4 4

Flow value = 10

10 13 1010 10 10X X X

Page 22: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

22

Towards an Algorithm

Find an s-t path where each arc has u(e) > f(e) and "augment" flow along the path.

Repeat until you get stuck. Greedy algorithm fails.

s

4

2

5

3 t 10 13 10

4

0 0

10 10 10

0

40

4 4

Flow value = 10

s

4

2

5

3 t 10 13 10

5

4 4

10 6 10

4

54

5 5

Flow value = 14

Page 23: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

23

Residual Arcs

Original graph G = (V, E). Flow f(e). Arc e = (v, w) E.

Residual graph: Gf = (V, Ef ). Residual arcs e = (v, w) and eR = (w, v). "Undo" flow sent.

v w 176

Capacity

Flow

v w 11

Residual capacity

6Residual capacity

Page 24: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

24

Residual Graph and Augmenting Paths

Residual graph: Gf = (V, Ef ). Ef = {e : f(e) < u(e)} {eR : f(e) > 0}.

s

4

2

5

3 t 10 13 10

4

0 0

10 10 10

0

40

4 4

s

4

2

5

3 t 10 10 10

4 4

4 4

3

Ee

Eeefeueu Rf iff(e)

if)()()(

G

Gf

Page 25: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

25

Augmenting Path

Augmenting path = path in residual graph.

s

4

2

5

3 t 10 13 10

4

0 0

10 10 10

0

40

4 4

s

4

2

5

3 t 10 10 10

4 4

4 4

3

G

Gf

4 4

6

4

4

X

X

X

X

X

Page 26: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

26

Augmenting Path

Augmenting path = path in residual graph. Max flow no augmenting paths ???

s

4

2

5

3 t 10 13 10

4

4 4

10 6 10

4

44

4 4

s

4

2

5

3 t 10 6 10

4 4

4 4

7

Flow value = 14

G

Gf

Page 27: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

27

Max-Flow Min-Cut Theorem

Augmenting path theorem (Ford-Fulkerson, 1956): A flow f is a max flow if and only if there are no augmenting paths.

MAX-FLOW MIN-CUT THEOREM (Ford-Fulkerson, 1956): the value of the max flow is equal to the value of the min cut.

We prove both simultaneously by showing the TFAE: (i) f is a max flow. (ii) There is no augmenting path relative to f. (iii) There exists a cut (S, T) such that |f| = cap(S, T).

Page 28: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

28

Proof of Max-Flow Min-Cut Theorem

We prove both simultaneously by showing the TFAE: (i) f is a max flow. (ii) There is no augmenting path relative to f. (iii) There exists a cut (S, T) such that |f| = cap(S, T).

(i) (ii)

We show contrapositive. Let f be a flow. If there exists an augmenting path, then we can

improve f by sending flow along path.

(iii) (i) Next slide.

(iii) (i) This was the Corollary to Lemma 2.

Page 29: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

29

Proof of Max-Flow Min-Cut Theorem

We prove both simultaneously by showing the TFAE: (i) f is a max flow. (ii) There is no augmenting path relative to f. (iii) There exists a cut (S, T) such that |f| = cap(S, T).

(ii) (iii) Let f be a flow with no augmenting paths. Let S be set of vertices reachable from s in residual graph.

– clearly s S, and t S by definition of f

T)(S,

)(

)()(

ofout

to in ofout

cap

eu

efeff

Se

SeSe

s

t

Original Network

S T

Page 30: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

30

Augmenting Path Algorithm

b bottleneck(P) FOREACH e P IF (e E) // forward arc f(e) f(e) + b ELSE // backwards arc f(eR) f(e) - bRETURN f

Augment (f, P)

FOREACH e E f(e) 0Gf residual graph

WHILE (there exists augmenting path P) f augment(f, P) update GfRETURN f

FordFulkerson (V, E, s, t)

Page 31: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

31

Running Time

Assumption: all capacities are integers between 0 and U.

Invariant: every flow value f(e) and every residual capacities uf (e) remains an integer throughout the algorithm.

Theorem: the algorithm terminates in at most | f * | nU iterations.

Corollary: if U = 1, then algorithm runs in O(mn) time.

Integrality theorem: if all arc capacities are integers, then there exists a max flow f for which every flow value f(e) is an integer.

Note: algorithm may not terminate on some pathological instances (with irrational capacities). Moreover, flow value may not even converge to correct answer.

Page 32: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

32

1000

0

s

4

2

t 1

100

100

0

0

0

Choosing Good Augmenting Paths

Use care when selecting augmenting paths.

100

Page 33: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

33

1000

0

s

4

2

t 1

100

100

0

0

0

Choosing Good Augmenting Paths

Use care when selecting augmenting paths.

100

1

1

1

X

X

X

Page 34: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

34

1001

0

s

4

2

t

100

1

1

0

Choosing Good Augmenting Paths

Use care when selecting augmenting paths.

100

100

1

Page 35: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

35

1001

0

s

4

2

t 1

100

100

1

1

0

Choosing Good Augmenting Paths

Use care when selecting augmenting paths.

100

1

0

1

X

X

X

Page 36: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

36

s

4

2

t 1

100

100 100

100

0

1

1 1

1

Choosing Good Augmenting Paths

Use care when selecting augmenting paths.

200 iterations possible.

Page 37: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

37

Choosing Good Augmenting Paths

Use care when selecting augmenting paths. Some choices lead to exponential algorithms. Clever choices lead to polynomial algorithms.

Goal: choose augmenting paths so that: Can find augmenting paths efficiently. Few iterations.

Edmonds-Karp (1972): choose augmenting path with Max bottleneck capacity. (fat path) Sufficiently large capacity. (capacity-scaling) Fewest number of arcs. (shortest path)

Page 38: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

38

Capacity Scaling

Intuition: choosing path with highest bottleneck capacity increases flow by max possible amount.

Don't worry about finding exact highest bottleneck path. Maintain scaling parameter . Let Gf () be the subgraph of the residual graph consisting of only

arcs with capacity at least .

110

s

4

2

t 1

170

102

122

Gf

110

s

4

2

t

170

102

122

Gf (100)

Page 39: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

39

Capacity Scaling

Intuition: choosing path with highest bottleneck capacity increases flow by max possible amount.

Don't worry about finding exact highest bottleneck path. Maintain scaling parameter . Let Gf () be the subgraph of the residual graph consisting of only

arcs with capacity at least .

FOREACH e E, f(e) 0 smallest power of 2 greater than or equal to U

WHILE ( 1) Gf() -residual graph WHILE (there exists augmenting path P in Gf()) f augment(f, P) update Gf() / 2 RETURN f

ScalingMaxFlow(V, E, s, t)

Page 40: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

40

Capacity Scaling: Analysis

L1. If all arc capacities are integers, then throughout the algorithm, all flow and residual capacity values remain integers.

Thus, = 1 Gf() = Gf , so upon termination f is a max flow.

L2. The outer while loop repeats 1 + log2 U times. Initially U < 2U, and decreases by a factor of 2 each iteration.

L3. Let f be the flow at the end of a -scaling phase. Then value of the maximum flow is at most | f | + m .

L4. There are at most 2m augmentations per scaling phase. Let f be the flow at the end of the previous scaling phase. L3 |f*| | f | + m (2). Each augmentation in a -phase increases | f | by at least .

Theorem. The algorithm runs in O(m2 log (2U) ) time.

Page 41: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

41

Capacity Scaling: Analysis

L3. Let f be the flow at the end of a -scaling phase. Then value of the maximum flow is at most |f| + m .

We show that at the end of a -phase, there exists a cut (S, T) such that cap(S, T) | f | + m .

Choose S to be the set of nodes reachable from s in Gf().– clearly s S, and t S by definition of S

mΔ-T)cap(S,

eu

eu

efeff

SeSeSe

SeSe

SeSe

to in ofout ofout

to in ofout

to in ofout

)(

))((

)()(

s

t

Original Network

S T

Page 42: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

42

Choosing Good Augmenting Paths

Use care when selecting augmenting paths. Some choices lead to exponential algorithms. Clever choices lead to polynomial algorithms.

Goal: choose augmenting paths so that: Can find augmenting paths efficiently. Few iterations.

Edmonds-Karp (1972): choose augmenting path with Max bottleneck capacity. (fat path) Sufficiently large capacity. (capacity-scaling) Fewest number of arcs. (shortest path)

Page 43: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

43

Shortest Augmenting Path

Intuition: choosing path via breadth first search. Easy to implement.

– may implement by coincidence! Finds augmenting path with fewest number of arcs.

FOREACH e E f(e) 0Gf residual graph

WHILE (there exists augmenting path) find such a path P by BFS f augment(f, P) update GfRETURN f

ShortestAugmentingPath(V, E, s, t)

Page 44: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

44

Shortest Augmenting Path: Overview of Analysis

L1. Throughout the algorithm, the length of the shortest path never decreases.

Proof ahead.

L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.

Proof ahead.

Theorem. The shortest augmenting path algorithm runs in O(m2n) time.

O(m+n) time to find shortest augmenting path via BFS. O(m) augmentations for paths of exactly k arcs. If there is an augmenting path, there is a simple one.

1 k < n O(mn) augmentations.

Page 45: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

45

Shortest Augmenting Path: Analysis

Level graph of (V, E, s). For each vertex v, define (v) to be the length (number of arcs) of

shortest path from s to v. LG = (V, EG) is subgraph of G that contains only those arcs

(v,w) E with (w) = (v) + 1.

s

2

3

5

6 t

s

2

3

5

6 t

= 0 = 1 = 2 = 3

LG:

G:

Page 46: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

46

Shortest Augmenting Path: Analysis

Level graph of (V, E, s). For each vertex v, define (v) to be the length (number of arcs) of

shortest path from s to v. L = (V, F) is subgraph of G that contains only those arcs

(v,w) E with (w) = (v) + 1. Compute in O(m+n) time using BFS, deleting back and side arcs. P is a shortest s-v path in G if and only if it is an s-v path L.

s

2

3

5

6 t

= 0

= 1

= 2

= 3

L:

Page 47: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

47

Shortest Augmenting Path: Analysis

L1. Throughout the algorithm, the length of the shortest path never decreases.

Let f and f' be flow before and after a shortest path augmentation. Let L and L' be level graphs of Gf and Gf '

Only back arcs added to Gf '

– path with back arc has length greater than previous length

s

2

3

5

6 t

= 0 = 1 = 2 = 3

L

s

2

3

5

6 t

L'

Page 48: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

48

Shortest Augmenting Path: Analysis

L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.

At least one arc (the bottleneck arc) is deleted from L after each augmentation.

No new arcs added to L until length of shortest path strictly increases.

s

2

3

5

6 t

= 0 = 1 = 2 = 3

L

s

2

3

5

6 t

L'

Page 49: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

49

Shortest Augmenting Path: Review of Analysis

L1. Throughout the algorithm, the length of the shortest path never decreases.

L2. After at most m shortest path augmentations, the length of the shortest augmenting path strictly increases.

Theorem. The shortest augmenting path algorithm runs in O(m2n) time.

O(m+n) time to find shortest augmenting path via BFS. O(m) augmentations for paths of exactly k arcs. O(mn) augmentations.

Note: (mn) augmentations necessary on some networks. Try to decrease time per augmentation instead. Dynamic trees O(mn log n) Sleator-Tarjan, 1983 Simple idea O(mn2)

Page 50: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

50

Shortest Augmenting Path: Improved Version

Two types of augmentations. Normal augmentation: length of shortest path doesn't change. Special augmentation: length of shortest path strictly increases.

L3. Group of normal augmentations takes O(mn) time. Explicitly maintain level graph - it changes by at most 2n arcs after

each normal augmentation. Start at s, advance along an arc in LG until reach t or get stuck.

– if reach t, augment and delete at least one arc– if get stuck, delete node

s

2

3

5

6 t

= 0 = 1 = 2 = 3

LG

Page 51: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

51

Shortest Augmenting Path: Improved Version

s

2

3

5

6 t

LG

s

2

3

5

6 t

LG

s

2 5

6 t

LG

augment

augment

delete 3

Page 52: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

52

Shortest Augmenting Path: Improved Version

s

2 5

6 t

LG

s

2 5

6 t

LG

STOP: length of shortest path must have strictly increased

augment

Page 53: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

53

Shortest Augmenting Path: Improved Version

ARRAY pred[v V]LG level graph of Gfv s, pred[v] nil

REPEAT WHILE (there exists (v,w) LG) pred[w] v, v w IF (v = t) P path defined by pred[] f augment(f, P) update LG v s, pred[v] nil delete v from LG UNTIL (v = s)

RETURN f

AdvanceRetreat(V, E, f, s, t)

advance

retreat

augment

Page 54: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

54

Shortest Augmenting Path: Improved Version

Two types of augmentations. Normal augmentation: length of shortest path doesn't change. Special augmentation: length of shortest path strictly increases.

L3. Group of normal augmentations takes O(mn) time. Explicitly maintain level graph - it changes by at most 2n arcs after

each normal augmentation. Start at s, advance along an arc in LG until reach t or get stuck.

– if reach t, augment and delete at least one arc– if get stuck, delete node– at most n advance steps before one of above events

Theorem. Algorithm runs in O(mn2) time. O(mn) time between special augmentations. At most n special augmentations.

Page 55: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

55

Choosing Good Augmenting Paths: Summary

First 4 rules assume arc capacities are between 0 and U.

Method AugmentationsAugmenting path nU

Max capacity m log UCapacity scaling m log U

Shortest path mn

Running timemnU

m log U (m + n log n)m2 log U

m2nImproved shortest path mn mn2

Improved capacity scaling m log U mn log U

Page 56: Taken from Kevin Wayne’s slides (Princeton University) COSC 3101A - Design and Analysis of Algorithms 13 Maximum Flow

56

History

Dantzig

Discoverer

Simplex

Method Big-Oh

mn2U1951

Year

Ford, Fulkerson Augmenting path mnU1955Edmonds-Karp Shortest path m2n1970

Dinitz Shortest path mn21970Edmonds-Karp, Dinitz Capacity scaling m2 log U1972

Dinitz-Gabow Capacity scaling mn log U1973Karzanov Preflow-push n31974

Sleator-Tarjan Dynamic trees mn log n1983Goldberg-Tarjan FIFO preflow-push mn log (n2 / m)1986

. . . . . . . . .. . .

Goldberg-Rao Length function m3/2 log (n2 / m) log Umn2/3 log (n2 / m) log U

1997