Max flow

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Max flow

• Jorsef

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Definitions
Capacity
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Sink
Source
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A flow
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A Cut
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Capacity/Flow Across A Cut
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More Definition Residual Graph
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• A path from source to sink in the residual graph
  of a given flow
• If there is an augmenting path in the residual
  graph, we can push more flow

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Ford-Fulkerson Method

• initialize total flow to 0
• residual graph G G
• while augmenting path exist in G
• pick a augmenting path P in G
• m bottleneck capacity of P
• add m to total flow
• push flow of m along P
• update G

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Example
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Example
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Example
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Example
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Example
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Example
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Example
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Answer Max Flow 4
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Answer Minimum Cut 4
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Find Augmenting Path

• initialize total flow to 0
• residual graph G G
• while augmenting path exist in G
• pick a augmenting path P in G
• m bottleneck capacity of P
• add m to total flow
• push flow of m along P
• update G

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Maximum Capacity Augmentation

• One way to select augmenting paths is to select
  one with maximum residual capacity.
• Maximum capacity augmenting paths can be found
  using a bottleneck version of Dijkstras algorithm

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Capacity Scaling Algorithm

• It allows augmenting paths to be found using
  breadth-first search, while still using large
  capacity paths first.
• The algorithm starts by letting D c where c is
  the maximum edge capacity. It then repeats the
  following step until D lt 1.
• find a shortest augmenting path in the subgraph
  RD , of the residual graph R, containing edges
  with residual capacity ? D
• add as much flow as possible to the path
• if there is no augmenting path, replace D by D/2

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Augmenting by Shortest Paths

• Another way to select augmenting paths is to
  augment along a path with the fewest edges. This
  yields an O(m2n) running time.

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Dinics Algorithm

• Dinics algorithm is based on the observation
  that many consecutive steps in the shortest path
  version of the augmenting path algorithm involve
  augmenting paths of the same length.
• We can avoid some redundant effort by ignoring
  edges that cannot possibly appear in augmenting
  paths of the current length.

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Dinics Algorithm

• Let G be a graph, f be a flow function on G and
  R the corresponding residual graph.
• let level(u) be defined as the number of edges in
  a shortest path from s to u in R
• only edges (u,v) for which level(v) level(u) 1
  can possibly be on a shortest augmenting path
• Dinics algorithm restricts its search for an
  augmenting paths to those edges for which (u,v)
  for which level(v) level(u) 1.

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Dinics Algorithm

• Dinic's algorithm begins by setting f(u,v) to
  zero for all edges in G, computes level(u) for
  all vertices u and then repeats the following
  step as long as level(t)ltn.
• while there is an augmenting path using edges
  (u,v) with level(v) level(u)1, select such a
  path and saturate it
• recalculate level(u) for all u

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Other algorithms

• Dinics Algorithm with Dynamic Trees
• Preflow-Push Method
• The FIFO Algorithm
• Highest Label Preflow Push algorithm
• Excess Scaling Algorithm

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Application Bipartite Matching

• Marriage Problem

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A Matching
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Maximum Matching
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Maximum Flow Problem
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Maximum Flow/Matching
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If we have many sources and sinks?
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RookAttack problem
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thank you