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ACS 7101/3 – Advanced Algorithm Design

Finding the maximum matching of a bipartite graph

Final Project

November 2008

Finding the maximum matching of a bipartite graph

From an initial matching

We want to find the maximum matching

1 2 3 4 5 6

1 2 3 4 5 6

7 8 9 10 11 12

7 8 9 10 11 12

- preconditions:
- The graph stored in a text file is represented by positive integer numbers.
- The nodes must be consecutive and no node should be represented with a number bigger than the total number of nodes.
- This code is designed for a balanced bipartite graph.

Notes:

- Since we are using positive integers to name each node, we are not going to
- use the index 0 of the array.

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

The general procedure is:

M1 G1 G1’ P M1 P = M2

M2 G2 G2’ P1 & P2 M2 P1 P2 = M3

In this example M3 is the maximum matching

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

We start with the graph stored in a text file as pairs u, v

Store the file in memory using an Array of Objects

Graph.txt

1 7

2 8

2 9

3 9

3 10

4 10

11

…

…

1 2 3 4 5 6

7 8 9 10 11 12

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

M1 G1

- count the nodes
- readFile.cpp
- randomMatching.cpp
- or exampleMatching.cpp
- initialize rest of the Array

G1

G1 is a directed graph

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

G1 G1’ : finding j*

- Level 0: (first half)
- if M = 0 => L = 0
- while j* != 0
- Level k odd (we want to go down on the directed graph)
- for the first half
- assign k to all the nodes minus the one pointed by pm in the list
- Level k even (we want to go up)
- for the second half
- assign k to all the matching node
- Note:
- when L = current level and M = 0 => j* = L

- findLevels.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

G1’ P

- Use a stack to keep track of the path
- Starting at Level 0:
- Push (i)
- Read the list while it doesn’t belong to a path (P = 0) AND is not pointed by pm
- while level(j) is greater than 0 and <= j*, AND it doesn\'t belong to a path AND it does belong to a matching: push(j); level ++;

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M1 P = M2

- If we found a free node then we found a path => empty the stack, set P to 1,
- while doing the symmetric difference and updating the matching…

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

Idea behind the symmetric difference:

3 edges in the path:

(1, 7)

(7, 6)

(6, 12)

1 2 3 4 5 6

1 2 3 4 5 6

M2

P1

7 8 9 10 11 12

7 8 9 10 11 12

Edge (1, 7)

In M2 doesn’t belong to a match match

(A[1].M = 0 AND A[7].M = 1)

M3

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

Idea behind the symmetric difference:

3 edges in the path:

(1, 7)

(7, 6)

(6, 12)

1 2 3 4 5 6

1 2 3 4 5 6

M2

P1

7 8 9 10 11 12

7 8 9 10 11 12

Edge (1, 7)

In M2 doesn’t belong to a match match

(A[1].M = 0 AND A[7].M = 1)

M3

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

Idea behind the symmetric difference:

3 edges in the path:

(1, 7)

(7, 6)

(6, 12)

1 2 3 4 5 6

1 2 3 4 5 6

M2

P1

7 8 9 10 11 12

7 8 9 10 11 12

Edge (7, 6)

Is a matching in M2 ignore

(A[7].M = 1 AND A[6].M = 1)

M3

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

Idea behind the symmetric difference:

3 edges in the path:

(1, 7)

(7, 6)

(6, 12)

1 2 3 4 5 6

1 2 3 4 5 6

M2

P1

7 8 9 10 11 12

7 8 9 10 11 12

Edge (6, 12)

In M2 doesn’t belong to a match match

(A[6].M = 1 AND A[12].M = 0)

M3

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

How does the code work:

We only evaluate alternate edges

Edge (1, 7)

In M2 doesn’t belong to a match match

(A[1].M = 0 AND A[7].M = 1)

A[1].M = 1 AND A[7].M = 1

…

…

…

Edge (6, 12)

In M2 doesn’t belong to a match match

(A[6].M = 1 AND A[12].M = 0)

A[6].M = 1 AND A[12].M = 1

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

P M2 P1 = M3

After repeating the process:

M3

1 2 3 4 5 6

7 8 9 10 11 12

- findPaths.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

writeGraph.txt

1 7 Match

2 8 Match

2 9

3 9

3 10 Match

…

7 1 Match

6

2 Match

…

The final step is to write al the pairs (u, v) back into a text file including the matching edges found

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

List of files coded:

main.cpp

linkedListBG.h

graph.txt

array.cpp

readFile.cpp

display.cpp

randomMatching.cpp

exampleMatching.cpp

findLevels.cpp

findPaths.cpp

writeFile.cpp

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

Improvements to be done before testing it with bigger graphs:

- Change “pos” to an auxiliary pointer
- Store repeated calls to linked list in a variable
- Check end of list (now is done with count() should be checking if the pointer points to null)
- error control:
- Open file
- List is empty
- …
- Finish the random initial matching
- …

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

Problems encounter during the process:

- The implementation was not clear in the book
- How to store the graph in memory
- How to mark the matching edges
- How to find the levels
- How to find the symmetric difference
- C++ language
- All the above plus
- First time working with array of objects ever
- Plus linked lists and pointers (everywhere)

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

Positive actions during the process :

- C++
- Hello world
- Books – exercises
- coding pseudo codes and assignment 2
- design
- Discussions with professor
- brainstorming the Structure
- writing the document

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

What is next:

- Fix everything mentioned before
- Create a function to generate a random graph
- Test the code in an incremental way starting maybe with 50 nodes and increment it up to 20.000 nodes

ACS 7101 – Advance Data Structures

November 2008

Finding the maximum matching of a bipartite graph

Questions?

ACS 7101 – Advance Data Structures

November 2008

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