A Graph Partitioning Algorithm by Node Separators

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

• 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
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
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
(No Transcript)
13
Bipartite graph matching

• 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.

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) lt Y

L0 f L1 b, c L2 e, g
d
a
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) lt Y

16
Finding an initial separator
17
Finding an initial separator

• Minimum degree algorithm J.A. George, 1981

4
4
4
4
1
3
4
3
4
5
2
6
3
5
3
3
4
3
3
4
3
4
18
Finding an initial separator

• Minimum degree algorithm J.A. George, 1981

4
4
4
4
3
4
3
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 maxj Ci lt n/2. Initial
  separator adjacency (Cj)

18
13
17
19
12
22
1
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
• H.N. Djidjev, 1982, On the problem of
  partitioning planar graphs. SIAM J. Algebraic
  Discrete Methods. 3, 229-240.
• R.J. Lipton and R.E. Tarjan, 1979, A separator
  theorem for planar graphs, SIAM J. Appl. Math.
  36, 177-189.
• J.W.H Liu, 1989, A graph partitioning algorithm
  by node separators. ACM Transactions on
  Mathematical Software, Vol. 15, No. 3, 198-219.
• B. W. Kernighan and R.M. Karp, 1970, An efficient
  heuristic procedure for partitioning graphs. Bell
  Syst. Tech. 49, 291-307.
• 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, 225-231.
• 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. 183-217.
• Sparse matrix ordering
• J.A. George, 1973, Nested dissection of a regular
  finite element mesh. SIAM J. Numer. Anal, 10,
  345-363.
• 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, 270-281.

25
The END