1. A Graph Partitioning Algorithm by Node Separators
Joseph W.H. Liu
ACM Transactions on Mathematical Software, 1989
Speaker: Andy A.K Jeng

2. Outline Problem definition - An example Applications
Iterative Separator Improvement Scheme
Bipartite graph matching
Finding an initial separator
Experimental results
Reference list

3. Introduction Definition – node separator
Given a connected and undirected graph G=(V,E), a node subset S is a node separator if the subgraph induced by the nodes in V but not in S has more than one connected component.

4. An Example – Node separator

5. An Example – Edge separator

6. The size of remaining connected graphs
Criteria of node separator
For different classes of problems, the size of what is regarded as good separators varies.
Any reasonable measurement involves
(1) the size of separator
(2) the size of the remaining connected components
Authors
The size of separator
The size of remaining connected graphs
Lipton and Tarjan,1979
≤ (8n)1/2
≤ 2n/3
Djidjev, 1982
≤ (6n)1/2
≤ (4n)1/2
可降低執行時間
增加平行運算效果

7. Applications
Its fundamental importance lies in its connection to the divide-and-conquer paradigm
Reordering for symmetric sparse matrices
Nested dissection algorithm George 1973
Node separator
Layout of VLSI circuits Leiserson 1980
Finding a construction to map graphs into layouts in the most area-efficient way
Edge separator

8. Reordering for symmetric sparse matrices
output
Gaussian Elimination needs O(np2) flops
Input
mapping
output
Nested dissection algorithm
Input

9. Iterative Separator Improvement Scheme

10. Iterative Separator Improvement Scheme

11. E2 E1
S - Y Y S
Questions:
(1)How to find a good initial separator S ?
(2)How to find a subset Y from separator S ?

12. | Adjacency (Y) | < | Y |
| Adjacent (Y) |

13. Bipartite graph matching
Maximum matching
Maximal matching
Nodes a, e and g are exposed nodes.
Augmenting path d a e b f c g

14. A matching M in a bipartite graph H is maximum if and only if there is no augmenting path in H with respect to M. C. Berge, 1957.
J. E. Hopcroft, 1973, provided an n2.5 algorithm to find maximum matchings in Bipartite graph. a b c e f g d
Maximum matching d a e b f c g
Maximal matching

15. Let x be an exposed node, if there is no augmenting path starting with x in H respect to M, then there exists a subset Y containing the node x such that | Adjacent (Y) | < | Y |
L0 = {f}
L1 = {b, c}
L2 = {e, g} d a
L0 + L2 = {e, f, g} = Y
L1 = {b, c} = adjacency (Y) e b f c g
Let M be a maximum matching of the bipartite graph H, if x is an exposed node in H respect to M, then there exists a subset Y containing the node x such that | Adjacent (Y) | < | Y |

16. Finding an initial separator

17. Finding an initial separator
Minimum degree algorithm J.A. George, 1981 4 2 4 2 4 4 1 3 4 2 3 4 5
(1) Mininum degree algorithm 原本使用在 reordering method for sparse matrix 2 2 6 3 2 5 3 3 4 3 3 4 3 4

18. Finding an initial separator
Minimum degree algorithm J.A. George, 1981 4 3 4 4 4 3 3 3 4 7 6 5 3 3 4 3 6 4 3 4

19. Finding an initial separator
xj is the node with the i-th degree order
Cj is a connected component in the subgraph {x1,x2,…,xj} that contains node xj
j = max{j : | Ci | < n/2} Initial separator = adjacency (Cj)
C15 18 13 17 19 12 1 22 3 4 5 16 7 2 14 11 9 20 8 6 15 10 20

20. Experimental results (1)
Initial separator
Iterative Improvement Scheme

21. Experimental results (2)
LS midlevel J.A. Geoger 1973
Iterative Improvement Scheme

22. Experimental results (3)
Iterative Improvement Scheme
LS midlevel + Initial separator

23. Reference list Graph separators Bipartite Graph matching
H.N. Djidjev, 1982, On the problem of partitioning planar graphs. SIAM J. Algebraic Discrete Methods. 3,
R.J. Lipton and R.E. Tarjan, 1979, A separator theorem for planar graphs, SIAM J. Appl. Math. 36,
J.W.H Liu, 1989, A graph partitioning algorithm by node separators. ACM Transactions on Mathematical Software, Vol. 15, No. 3,
B. W. Kernighan and R.M. Karp, 1970, An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. 49,
Bipartite Graph matching
J.E. Hopcroft and R.M Karp, 1973, An n**(5/2) algorithm for maximum matching in bipartite graphs. SIAM J. Computer. 2,
C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Prentice-Hill, Englewood Cliffs, N.J.

24. References list (cont. )
Minimum degree algorithm
J.A. George and J.W.H. Liu, The evolution of the minimum degree ordering algorithm. SIAM Rev. 31, 1-19.
D.J. Rose . A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations . In Graph Theory and Computing, R. Read, Ed. Academic Press, Orlando, Fla
Sparse matrix ordering
J.A. George, 1973, Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal, 10,
J.A. George and J.W. H. Liu, Computer solution of Large Sparse positive Definite Systems Prentice-Hall, Englewood Cliffs, N.J
VLSI layout
C. Leiserson, 1980, An efficient graph layout for VLSI. In Proceeding of the 21st Annual Symposium on the Foundations of Computer Science, IEEE, New York,

25. The END