Hiren patel (hkp4027)

1

Network Flow

• Hiren patel (hkp4027)
• Ujjval patel (ujp4126)

2
Problem Defination

• Implement the Ford-Fulkerson
• algorithm for computing network flow in
• bipartite graphs.

3
Ford Fulkerson Algorithm

• Initialize flow f to 0
• While there exists an augment path p
• do augmant flow f along path p
• Return f

4
Maximum Bipartite Matching

• Given two sets of vertices Group1 and Group2 ,
• And edges between the nodes of Group1 to
  nodes of Group2,
• find the matching with maximum cardinality.
• How ford fulkerson algorithm can be used in
  matching problem?

5
Implementation

• PLATFORM Microsoft windows 9x,NT,2000
• PROGAMMAING LANGUAGE Visual C .NET
• PROGRAMMING TOOL Microsoft Visual Studio .NET
  2003
• Code implemented woks in two modes
• Ford Fulkerson mode
• Bipartite mode

6

• DFS Algorithm is used for finding a source to
  sink path.
• Data structures used
• Adjacency matrix representation for the graph.
• bInput N by N Boolean array to store graph.
• iCapacity N by N int32 array to store
  capacities of edges.
• iFlow N by N int32 array to store flow from
  the edges.
• All data structures used are instantiated
  dynamically .

7
Analysis

• Running Time Analysis
• O (E f) , where f is the maximum
  flow and E is no of edges in network.
• Space Complexity
• O (N N), where N is the number of nodes in
  the graph.

8
Performance of the Ford Fulkerson over different
inputs.
Nodes Edges Flow Augmented paths Ideal Complexity O( E f) Actual complexity Space Complexity O(NN)
4 5 15 3 75 21 16
8 11 20 4 220 55 64
20 24 30 3 49 750 400
9
Performance of the Max. Bipartite matching over
different inputs.
Nodes Edges Flow Augmented paths Ideal Complexity O( E f) Actual complexity Space Complexity O(NN)
10 17 5 5 50 49 100
16 28 8 8 128 78 256
20 34 10 10 200 90 400
26 52 12 12 288 128 676