1. Computing Maximum Non-Crossing Matching in Convex Bipartite Graphs
Danny Z. Chen, Xiaomin Liu, and Haitao Wang
University of Notre Dame
Indiana, USA
FAW-AAIM 2012, Beijing, China

2. Bipartite Graphs

3. Convex Bipartite Graphs
Each red vertex connects to a subset of consecutive blue vertices
The blue vertices have an order

4. Maximum Matching in Graphs
A matching is sub-set of graph edges such that no two edges connect to the same vertex

5. Maximum Non-Crossing Matching (MNCM)
Suppose the graph embedding is given
Each edge is a line segment
Any two edges in the matching do not cross each other

6. Problem Definition
Given a convex bipartite graph of n vertices and m edges G
Represented implicitly in O(n) space, despite m=ϴ(n2)
Give the top and bottom edges of each red vertex
Goal: find a maximum non-crossing matching in G
top edge
bottom edge

7. Motivation A sub-problem in our study of medical imaging
Applications in VLSI design

8. Previous Work
Proposed by Kajitami and Takahashi, 86’ (Symp. On Circuits and Systems)
O(n2) time algorithm
Reduce it to the problem of finding the longest increasing subsequence in a permutation of size m
O(mlog n) time, Fredman 75’, Widmayer and Wong, 82’
Malucelli, Ottamann, and Pretolani, 93’ (Discrete Applied Mathmatics)
A labeling algorithm
O(mloglog n) time for general bipartite graphs
O(m+nlog n) time for convex bipartite graphs

9. Our Result Convex bipartite graphs The approaches O(nlog n) time
Based on the labeling algorithm
New observations
A new data structure

10. Notation For any two edges in a non-crossing matching
One edge is always above the other e1 e1 e2 e2
e1 is above e2
e1 is NOT above e2

11. Partial Maximum Non-Crossing Matching
Each edge e defines a sub-graph consisting of all edges of G above e as well as e
An MNCM in the sub-graph is a partial MNCM for e
An algorithmic scheme:
Consider the red vertices
incrementally from top to
bottom, and determine
the corresponding partial MNCM e

12. An intuition

13. The Labeling Algorithm (Previous Work)
A label for each edge e: the size of the partial MNCM of e
A label for each blue vertex: the largest label of the incident edges of the blue vertex 1 1
Rule: the label value of the edge equals
one plus the largest label of the blue
vertices above u 2 2 u 2 3 3 2 3
O(mlog n) time: priority search tree 3

14. A Different Implementation for Storing the Labels (Previous Work)
For each label value, we only need to know the highest blue vertex with the same label value
A map M: M[i] refers to the highest blue vertex with label i 1
M[1]
The values in M are increasing 1
M[2] 2 2 u 3
O(mlog n) time: binary search
O(mloglog n) time:
integer data structure 2
M[3] 3 3 2 3 4 4
M[4] 3

15. A New Observation for Updating M on Convex Bipartite Graphs (Our Work)
M[i]
i+1
top edge
M[i+1]
i+2 v
M[i+2]
i+3
M[i+3]
i+4
bottom edge
M[i+4]

16. A New Observation for Updating C (cont.)
M[i] …
M[i+1] …
M[i+1]
Update M
M[i+2]=1+M[i+1] … v ……
M[h] …
M[h-1]=1+M[h-2]
M[h]=1+M[h-1] …
M[h+1]
M[h+1]=1+M[h]
a range-shift operation

17. A New Data Structure Storing the elements in M
Implementing the range-shift operations
O(log n) time
The segment tree does not work
Because insertions and deletions are involved in the range-shift operations
Overall running time of the algorithm: O(nlog n)

18. Finding an Actual MNCM Our algorithm computes the map M
With M, find an actual MNCM
Previous work: O(m) time, m=ϴ(n2)
Our new result: O(n) time, a greedy algorithm
The key: one particular red vertex
The lowest red vertex in an MNCM
The map M is not needed!

19. An example

20. Conclusion
A new algorithm for computing MNCM in convex bipartite graphs
O(nlog n) time
Previous work: O(m+nlog n), m=O(n2)
Open problem:
A maximum matching can be found in O(n) time
Can we find an MNCM in O(n) time?
Determining the particular red vertex looks easier?

21. Thank you