Introduction to Algorithms

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Introduction to Algorithms

• Maximum Flow
• My T. Thai _at_ UF

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Flow networks

• A flow network G (V, E) is a directed graph
• Each edge (u, v) ? E has a nonnegative capacity
• If (u, v) ? E , then
• If ,
• A source s and a sink t

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Flow networks

• A flow in G is a real-valued function
  that satisfies the two properties
• Capacity constraint
• ?u, v?V 0 ? f(u, v) ? c(u, v)
• Flow conservation
• ?u?V ? s, t ?v ?V f(u, v) ?v ?V f(v, u)
• The value f of a flow f

(the total flow out of the source minus the flow
into the source)
Label of each edge f/c
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Max-Flow Problem

• Input A flow network G (V, E), s, and t
• Output Find a flow of maximum value
• Applications
• Flow of liquid through pipes
• Information through communication networks
• Flow in a transport network

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Remove antiparallel edges
(v1, v2) and (v2, v1) are antiparallel edges
Vertex v is added to remove edge (v1, v2)
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Multiple sources and sinks
Add supersource s
Add supersink t
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Ford-Fulkerson Method
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Example
Augmenting path sv2v3t
All flows in the augmenting path are increased by
4 units
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Residual network

• Given flow f, the residual capacity of edge (u,
  v)
• Residual network induced by f is Gf (V, Ef)

?
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Flow augmentation

• f is a flow in G and f is a flow in Gf , the
  augmentation of flow f by f( ), is a
  function from , defined by

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Augmenting path

• Any simple path p from s to t in G f is an
  augmenting path in G with respect to f
• Flows on edges of p can be increased by the
  residual capacity of p

Cf(p) 4 with p is the shaded path
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Increase the value of flow along augmenting paths
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Cuts

• A cut (S,T) of a flow network G (V, E) is a
  partition of V into S and T V ? S such that s ?
  S and t ? T.
• The net flow across the cut
• The capacity of the cut
• A minimum cut of G is a cut
• whose capacity is minimum
• over all cuts of G

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• Proof
• Flow conservation gt
  ,

gt
(regroup the formula)
(partition V to S and T)
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Upper bound of flow value

• Proof
• Let (S, T) be any cut, f be any flow

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Max-flow min-cut theorem

• Proof
• (1) gt (2) If Gf has an augmenting path p, then
  by Corollary 26.3, there exists a flow of value
  f Cf(p) gt f that is a flow in G
• (2) gt (3) Suppose (2) holds. Define S v?V
  ? path from s to v in Gf, and T V S. (S, T)
  is a cut. For u ? S, v ? T, f(u,v) c(u,v)
• By Lemma 26.4, f f(S,T) c(S,T)
• (3) gt (1) By Corollary 26.5, f c(S,T)
  implies f is a maximum flow

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Ford-Fulkerson algorithm

• Running time
• If capacities are integers, then O(Ef), where
  f is the maximum flow
• With irrational capacities, might never
  terminate.

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Example
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(No Transcript)
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No augmenting path
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Bad example

• FORD-FULKERSON runs 1000,000 iterations

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Edmonds-Karp algorithm

• Do FORD-FULKERSON, but compute augmenting paths
  by BFS of Gf . Augmenting paths are shortest
  paths from s to t in Gf , with all edge weights
  1
• Running time O(VE2)
• There are O(VE) flow augmentations
• Time for each augmentation (BFS) is O(E)

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Analysis of E-K

• Proof (by contradiction)
• Suppose that there exist 2 continuous flows f and
  f such that ?f?(s,v) ? ?f(s,v) for some v ? V
• W.l.o.g suppose v is the nearest vertex (to s)
  satisfying ?f?(s,v) lt ?f(s,v) ()

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Proof of Lemma 26.7

• Let u be the vertex before v on the shortest path
  from s to v
• gt
• gt u is not the nearest vertex to s satisfying
  ()
• gt
• Claim . If not we have

(Contradicted)
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• gt The augmentation must increase flow v to u
• Edmonds-Karp augments along shortest paths, the
  shortest path s to u in Gf has v ? u as its last
  edge

(Contradicted)
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Time bound

• Proof
• Suppose p is augmenting path, (u, v) ? p, and
  cf(p) cf(u,v). Then (u, v) is critical, and
  disappears from the residual network after
  augmentation.
• Before (u, v) becomes a critical edge again,
  there must be a flow back from v to u

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Time bound

• Claim each of the E edges can become critical
  V /2 - 1 times
• Consider edge (u, v), when (u, v) becomes
  critical first time, df(s, v) df(s, u) 1
• (u, v) reappears in the residual network after
  (v, u) is on an augmenting path in Gf , df(s,
  u) df(s, v) 1
• gt

gtThe claim is proved gt The theorem is proved
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Maximum bipartite matching

• A matching is a subset of edges M ? E such that
  for all v ? V, 1 edge of M is incident on v.
• Problem Given a bipartite graph find a matching
  of maximum cardinality.

each edge has capacity 1
(A maximum matching)
(Reduce to the maximum flow problem)
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Integrality theorem
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Summary

• Max-flow min-cut theorem the capacity of the
  minimum cut equals to the value of the maximum
  flow
• Ford-Fulkerson method iteratively find an
  augmenting path and augment flow along the path
• Ford-Fulkerson algorithm find an arbitrary
  augmenting path in each iteration
• If capacities are integers, then O(Ef), where
  f is the maximum flow
• With irrational capacities, might never terminate.

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Summary

• Edmonds-Karp algorithm find a shortest
  augmenting path in each iteration
• Time O(VE2)
• Integrality theorem if all capacities are
  integral values, then the value of the maximum
  flow as well as flows on edges are integers