variants in which the number of nodes is forced to be exactly half that of the original sub-

domain. The second measure is more dif¬cult to control. Thus, the top part of Figure 13.9

shows the result of the RGB algorithm on a sample ¬nite-element mesh. This is a vertex-

based partitioning. The dashed lines are the cut-edges that link two different domains.

An approach that is competitive with the one described above is that of double striping.

This method uses two parameters , such that . The original graph is ¬rst ¨ y y ¨ y y

c c D

partitioned into large partitions, using one-way partitioning, then each of these partitions

y

c

is subdivided into partitions similarly. One-way partitioning into subgraphs consists ¨y y

of performing a level set traversal from a pseudo-peripheral node and assigning each set of

roughly consecutive nodes in the traversal to a different subgraph. The result of this

IEi

y

approach with is shown in Figure 13.9 on the same graph as before. As can

¨ y D'y

c D

be observed, the subregions obtained by both methods have elongated and twisted shapes.

This has the effect of giving a larger number of cut-edges.

k

q p z7 w—w

{ { {7 #

¢ 4 ¡ ¡

£¤§ S

¡

™

% # 3¡

!

The RGB algorithm (top) and the double-

striping algorithm (bottom) for partitioning a graph into 16

subgraphs.

‘

—t

7 u pU¢ 7 p z7 yw— y w u

{ {˜p p { p

˜

¡

“£§

¢

! hC §¤ !

¤ …S !

§

¥

¤ ¤

There are a number of heuristic ways to remedy this. One strategy is based on the

fact that a level set traversal from nodes can be de¬ned instead of only one node. These

nodes are called the centers or sites. Each subdomain will expand from one of these

centers and the expansion will stop when it is no longer possible to acquire another point

that is not already assigned. The boundaries of each domain that are formed this way will

tend to be more “circular.” To smooth the boundaries of an initial partition, ¬nd some center

point of each domain and perform a level set expansion from the set of points. The process

can be repeated a few times.

vx¤ … ¢

¡¦ ™ 24 ) 6 ¢'

§8 ¡¥¥ 7 2 9@20 (§# (& ¢ §8 5 '2 £¨ # % #

64 ¢

'

! 0¦ 974

3 86

¨©§¥

¦¤

£

£

1. Find a partition . i¢ii’¨

GD

F tc thhht a

2. For Do: W

'S˜ X w

DY ti¢i¢’e

i hhht S˜

Y

X

3. For Do:

D‚ 2hi¢hit e

th

}

3 p 3 x u˜ z 7R

4. Find a center of . Set . 4

p p R D

5. EndDo

¨ 3 uc 3 F 3

6. Do a level set traversal from . Label each child t ¢ii¢t

thhh a

7. in the traversal with the same label as its parent.

W

8. For set := subgraph of all nodes having label

p

D‚ 2hi¢hit e

th

9. EndDo

™

# & $"

%#!

Multinode expansion starting with the parti-

tion obtained in Figure 13.9.

‘

˜ ˜ ‚ —

{ ˜ wU

7p #

¡

§

x"

!

For this method, a total number of cut-edges equal to 548 and a rather small standard

deviation of 0.5 are obtained for the example seen earlier.

Still to be decided is how to select the center nodes mentioned in line 4 of the al-

gorithm. Once more, the pseudo-peripheral algorithm will be helpful. Find a pseudo-

peripheral node, then do a traversal from it until about one-half of the nodes have been

traversed. Then, traverse the latest level set (typically a line or a very narrow graph), and

take the middle point as the center.

A typical number of outer steps, nouter, to be used in line 2, is less than ¬ve. This

heuristic works well in spite of its simplicity. For example, if this is applied to the graph £

X i

obtained from the RGB algorithm, with , the partition shown in Figure 13.10 QS˜