1. Finding the maximum matching of a bipartite graph
ACS 7101/3 – Advanced Algorithm Design
Finding the maximum matching of a bipartite graph
Final Project
November 2008

2. Finding the maximum matching of a bipartite graph
From an initial matching
We want to find the maximum matching
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

3. Finding the maximum matching of a bipartite graph
The general procedure is:
M1 G1 G1’ P M1 P = M2
M2 G2 G2’ P1 & P M2 P1 P2 = M3
In this example M3 is the maximum matching
ACS 7101 – Advance Data Structures
November 2008

4. 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 …
ACS 7101 – Advance Data Structures
November 2008

5. 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

6. 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

7. 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 ++;
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

8. 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

9. 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)‏ M2 P1
Edge (1, 7)‏
In M2 doesn’t belong to a match match
(A[1].M = 0 AND A[7].M = 1)‏ M3
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

10. 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)‏ M2 P1
Edge (1, 7)‏
In M2 doesn’t belong to a match match
(A[1].M = 0 AND A[7].M = 1)‏ M3
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

11. 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)‏ M2 P1
Edge (7, 6)‏
Is a matching in M ignore
(A[7].M = 1 AND A[6].M = 1)‏ M3
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

12. 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)‏ M2 P1
Edge (6, 12)‏
In M2 doesn’t belong to a match match
(A[6].M = 1 AND A[12].M = 0)‏ M3
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

13. 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

14. Finding the maximum matching of a bipartite graph
P M2 P1 = M3
After repeating the process: M3
findPaths.cpp
ACS 7101 – Advance Data Structures
November 2008

15. Finding the maximum matching of a bipartite graph
writeGraph.txt
1 7 Match
2 8 Match
2 9
3 9
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

16. 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

17. 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

18. 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

19. 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

20. 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 nodes
ACS 7101 – Advance Data Structures
November 2008

21. Finding the maximum matching of a bipartite graph
Questions?
ACS 7101 – Advance Data Structures
November 2008