4 5 6

7 9

8

10 12 11

13 14

15

¥ ¢

D T ©

¡ Reverse Cuthill-McKee ordering starting with

0

.

Independent set orderings. The matrices that arise in the model ¬nite element prob-

lems seen in Figures 2.7, 2.10, and 3.2 are all characterized by an upper-left block that is

diagonal, i.e., they have the structure

‘t

–

¢

7 ¨ ¢ ¢¥

¡

– ¡

in which is diagonal and , and are sparse matrices. The upper-diagonal block

¢

7

corresponds to unknowns from the previous levels of re¬nement and its presence is due to

the ordering of the equations in use. As new vertices are created in the re¬ned grid, they

are given new numbers and the initial numbering of the vertices is unchanged. Since the

old connected vertices are “cut” by new ones, they are no longer related by equations. Sets

such as these are called independent sets. Independent sets are especially useful in parallel

computing, for implementing both direct and iterative methods.

¨B xB

©

¨¦

Referring to the adjacency graph of the matrix, and denoting by the ¢

$

C C

¨

¦ ¨

edge from vertex to vertex , an independent set is a subset of the vertex set such

that

x B7

yB

¤! (

¦ ¨© ¦ "

¦¨ ¨

if then or

¢ ¢

C C

ud

¤§ ¥ §¢

¡

¡© © ¢ ¡ ¥ ¥ © ¡ ¥

§

To explain this in words: Elements of are not allowed to be connected to other elements

of either by incoming or outgoing edges. An independent set is maximal if it cannot be

augmented by elements in its complement to form a larger independent set. Note that a

maximal independent set is by no means the largest possible independent set that can be

found. In fact, ¬nding the independent set of maximum cardinal is -hard [132]. In the

following, the term independent set always refers to maximal independent set.

There are a number of simple and inexpensive heuristics for ¬nding large maximal

independent sets. A greedy heuristic traverses the nodes in a given order, and if a node is

not already marked, it selects the node as a new member of . Then this node is marked

¦

along with its nearest neighbors. Here, a nearest neighbor of a node means any node

¦

linked to by an incoming or an outgoing edge.

V¤ G ¥ ¤¢

¡¦£ ¨# & T

B © § ¡DB

$ !8 £! "% !& §©

8

¢ £)

¨

§

QW F ©

}

1. Set .

˜ i)11zC Yy

2. For Do:

3. If node is not marked then

5 ¤¥ !

4.

5. Mark and all its nearest neighbors

6. EndIf

7. EndDo

˜ i)1)uCY ˜ 51)1u`‘ P1)1) 0 )

In the above algorithm, the nodes are traversed in the natural order , but they

‘ ! !

can also be traversed in any permutation of . Since the size of the

˜3

reduced system is , it is reasonable to try to maximize the size of in order to obtain

a small reduced system. It is possible to give a rough idea of the size of . Assume that the

maximum degree of each node does not exceed . Whenever the above algorithm accepts ¦

a node as a new member of , it potentially puts all its nearest neighbors, i.e., at most ¦

nodes, in the complement of . Therefore, if is the size of , the size of its complement,

¤

˜ ˜

3 3

, is such that , and as a result, §#

¦

¤ ¤ ¤

¨˜ — (

¤

¦

¢¦