emis 8374 max-flow in undirected networks updated 18 march 2008
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EMIS 8374
Max-Flow in Undirected Networks
Updated 18 March 2008
Slide 2
Max Flow in Undirected Networks
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s t
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Slide 3
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Replace Edge {i,j} With Arcs (i,j) and (j,i)
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Slide 4
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Max Flow in Directed Network
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Slide 5
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Max Flow in Directed Network
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Slide 6
Remove Bi-directional flows
if xij xji then xij = xij – xji and xji = 0
else xji = xji – xij and xij = 0
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2-2=0 4-2=2
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Slide 7
Max Flow in Undirected Network
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Arrows indicate flow direction
Slide 8
Remove Saturated Edges
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s t
S = {s, 4}T = {1, 2, 4, t}
Slide 9
Undirected s-t Cut
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s t
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u[S, T] = 24
Slide 10
All-Pairs Minimum Cut Problem
• Find the minimum value of u[A, B] where [A, B] is an partition of the nodes such that |A|>0 and |B|>0.
• Also known as the minimum 2-cut.
• Note that no specific source or sink nodes are specified.
Slide 11
Min 2-Cut Algorithm
• Since the network is undirected, u[A, B] = u[A, B]
• Don’t need to try s = j and t = i if we’ve already tried s = i and t = j
Slide 12
Min 2-Cut Algorithmv* = ;for s = 1 .. |N| - 1 for t = s + 1 .. |N| begin solve max s-t flow problem;
identify min cut [S, T]; if u[S, T] < v* then begin A = S; B = T; end end
Slide 13
Min 2-Cut Example
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Slide 14
Minimum Cut: s = 1, t = 2
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S = {1}T = {2, 3, 4, 5}u[S, T] = 17
Slide 15
Minimum Cut s = 1, t = 310
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S = {1,2}T = {3,4,5}u[S, T]=14
Slide 16
Minimum Cut: s = 1, t = 4
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S = {1,2,3,5}T={4}u[S, T]=12
Slide 17
Minimum Cut: s = 1, t = 510
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S = {1,2}T = {3,4,5}u[S, T]=14
Slide 18
Minimum Cut: s = 2, t =310
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S = {2,1}T = {3,4,5}u[S, T]=14
Slide 19
Observation• Suppose s = 2 and t = 3 and let [A, B] be a
minimum 2-3 cut.• Case 1: node 1 is in A
– [A, B] is also a 1-3 cut– Thus, we already know u[A, B] 14
• Case 2: node 1 is in N2
– [A, B] is also a 2-1 (1-2) cut– Thus, we already know u[A, B] 17
• There is no need to solve the max-flow problem for s = 2 and t =3.
Slide 20
An Improved Min 2-Cut Algorithm
• Consider a minimum 2-cut [A, B]• Let A be the set containing node 1.• Since |B| > 0, it must contain at least one
node in {2, 3, 4, 5}.• Thus we can discover [A, B] by solving only
|N| - 1 max flow problems with s =1 and t = 2, t = 3, …, t = |N|.
• Complexity is O(n f(n, m)) where f(n, m) is the complexity of solving a max flow problem
Slide 21
Minimum 2-Cut
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u[A, B]=12
A = {1,2,3,5}
B = {4}
Edge Connectivity
• In a so-called unweighted graph where each edge as a capacity of 1 unit, the capacity of a minimum 2-cut is known as the edge connectivity of the graph
• Connectivity is an important measure of a network’s reliability.
• In a telecommunications network an edge connectivity of two (2) means that the network can survive single-link failures.
Slide 22
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