Flow Networks

Ching-Chen HuangHsi-Yue Hsiao

CONTENTS

Network flows on directed acyclic graphs

Ford-fulkerson Algorithms -Residual networks

Network flows on directed acyclic graphsNetwork flows on directed acyclic graphs

DAG is a directed graph with no cycles. Flow network: a flow network G = (V,E) is a directed graph in which each edge (u,v) ∈ E has a nonnegative capacity c(u,v) >= 0. if (u,v) E , we assume that c(u,v) = 0.∉ We distinguish two vertices in flow network , source ss and sink tt. The source produces the material at a steady rate . The sink consumes the material at a steady rate.

Objective : how much of material can be imposed through the network that it can handle.

Flow: A flow in G is a real-valued function f : V × V → R that satisfies three properties:

1. Capacity constraint : For all u,v ∈ V, we require f(u,v) ≤ c(u,v). The net flow from one vertex to another must not exceed the given c

apacity. 2. Skew symmetry : For all u,v ∈ V, we require f(u,v) = -f(v,u). The net flow from a vertex u to a vertex v is the negative of the net fl

ow in the 3. Flow conservation : For all u ∈ V - {s,t}, we require ∑ f(u,v) = 0. v ∈ V The quantity f(u,v),which can be positive or negative , is called the n

et flow from vertex u to v. The value of the flow is defined as | f | = ∑ f(s,v) that is the totat net v ∈ V

Flow out of the source.

Residual Capacity Given a flow f in network G = (V, E). Consider a pair of vertices u, v ∈ V. Residual capacity:

The amount of additional flow we can push directly from u to v.

cf (u, v) = c(u, v) f (u, v) ≥ 0 (since f (u, v) ≤ c(u, v))

Example: c(u,v) = 16, f(u,v) = 5 cf (u, v) = 11

Residual Network

Residual network: Gf = (V, Ef ),Ef = {(u, v) ∈ V × V : cf (u, v) > 0} .

Each edge of the residual network can admit a positive flow.

Given flows f1 and f2, the flow sum f1 + f2 is (f1 + f2)(u, v) = f1(u, v) + f2(u, v)

for all u, v ∈ V.

Consider the network G=(V,E) shown in the figure below. Eachedge (u,v) ∈ E in the network is labeled with its capacity c(u,v).

s

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Flow: 1

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Residual graph

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Residual graph

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Max Flow : 5

Ford-Fulkerson Algorithm The Ford_Fulkerson method is iterative,

Starts with f(u,v) for (u,v) ∈ V, initial flow of value 0.

The method is based on the augmenting path which is

defined as a path from s to t along which we can push

more flow and then augment flow along this path.

Procedure Ford_Fulkerson_method(G,s,t): For each edge (u,v) ∈ E[G]

do f[u,v] 0 f[v,u] 0 While there exists a path p from s to t in the resi

dual network G1 do c1(p) min{ c1(u,v):(u,v) is in p} for each edge (u,v) in p do f[u,v] f[u,v] + c1(p) f[u,v] -f[u,v]

Running time for this algorithms is O(E * |f*|) where f* is the maximum flow found by algorithm.

First three Lines take time θ(E).

The while loop of last five lines is executed at most |f*| times, since the flow value increases by at least one unit in each iteration.

Ford-Fulkerson Algorithm

Ford-Fulkerson(G, s, t) ; G = (V, E) 1 for each edge (u,v) in E 2 f(u,v) = f(v,u) = 0 3 while exists path p from s to t in residual network Gf 4 do cf (p) = min{cf (u, v) : (u, v) is in p}

5 for each edge (u,v) on p 6 f(u,v) = f(u,v) + cf (p) 7 f(v,u) = -f(u,v)

s t

v1 v2

v4v313

16

1220

144

10 4 9

s t

v1 v2

v4v313

4/16

4/1220

4/144/4

10 4 4/9

7

7

Example:

s t

v1 v2

v4v313

12

820

104

10 4 5

474

4

s t

v1 v2

v4v313

11/16 7/20

11/144/4

7/10 4 4/97/7

4/12

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s t

v1 v2

v4v313

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813

34

3 11 5

4 711

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7

s t

v1 v2

v4v38/13

11/16 15/20

11/144/4

10 1/4 4/97/7

12/12

s t

v1 v2

v4v3

8

512

5

34

11 3 5

4 711 15

5

11

s

v1 v2

v4v312/13

11/16

11/144/4

10 1/4 97/7

12/12

t

19/20

s t

v1 v2

v4v31

512

1

3 4

11 3 97

11 19

12

11

Reference

Class Notes from Dr. Istvan Jonyer of Oklahoma State University

Class Notes from CSE 5311 of 2004, made by Hiren Patel, Ujjval Patel