1. Lecture 4: Matching Algorithms
String Matching
Pairwise Matching in Graphs

2. Knuth-Morris-Pratt
Automaton searching for the substring “ababcb”

3. Knuth-Morris-Pratt Algorithm

4. Boyer-Moore Algorithm

5. Boyer-Moore
Boyer-Moore searching for the substring “EDGEKIT”.

6. Bipartite Matching
The pairwise matching of members of a bipartite graph is another type of matching.
You are searching for a maximal matching (i.e. max number of pairings).

7. The Augmenting Path Algorithm
Given a bipartite graph Gn,m we are to find a maximal matching (max number of pairs) between the n nodes of group I and the m nodes of group II.  There is a greedy algorithm for the maximal matching problem:

8. Augmenting Path: An Example
Start with a bipartite graph.
Choose arbitrary pairings until no additional matches are possible. In this example nodes C, R and S are not matched.      
Matching edges are shown in bold

9. We will now build an augmenting path starting from node S. 
S-A=P-C. 
We exchange the matched and unmatched edges in this augmenting path increasing the total number of matches by one.      
A is matched to S 
B is matched to Q 
C is matched to P 
This is a maximal matching because there are no more unmatched nodes in one of the two groups.

10. Maximal Matching in a General Graph

11. For General Graphs
The Augmenting Path Algorithm does not necessarily produce a maximal matching in a general graph. In an odd cycle there will be two unmatched edges that share a vertex. The our DFS is started from such a vertex and the cycle is traversed the wrong way we will miss the augmenting path.