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There are some interesting fundamental properties of such matrices. Assuming the graph

is undirected, the matrix is symmetric. It can easily be seen that it is also negative semi

de¬nite (see Exercise 9). Zero is an eigenvalue and it is the smallest one. An eigenvector

associated with this eigenvalue is any constant vector, and this eigenvector bears little in-

terest. However, the second smallest eigenvector, called the Fiedler vector, has the useful

property that the signs of its components divide the domain into roughly two equal subdo-

mains. To be more accurate, the Recursive Spectral Bisection (RSB) algorithm consists of

sorting the components of the eigenvector and assigning the ¬rst half of the sorted vertices

to the ¬rst subdomain and the second half to the second subdomain. The two subdomains

are then partitioned in two recursively, until a desirable number of domains is reached.

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1. Compute the Fiedler vector of the graph . ¥

2. Sort the components of , e.g., increasingly. ¥

3. Assign ¬rst nodes to , and the rest to . ¨”

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4. Apply RSB recursively to , , until the desired number of partitions ” c ”

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The main theoretical property that is exploited here is that the differences between

the components of the Fiedler vector represent some sort of distance between the corre-

sponding nodes. Thus, if these components are sorted they would be grouping effectively

the associated node by preserving nearness. In addition, another interesting fact is that the

algorithm will also tend to minimize the number of cut-edges, i.e., the number of edges i ¢

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such that and . Let be a partition vector whose components are ¨” )

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Ideally, the objective function should be minimized subject to the constraint that

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. Note that here is a vector of signs. If, instead, the objective function

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lution would be the Fiedler vector, since is the eigenvector associated with the eigenvalue ˜

zero. The Fiedler vector is an eigenvector associated with the second smallest eigenvalue

of . This eigenvector can be computed by the Lanczos algorithm or any other method ef-

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¬cient for large sparse matrices. Recursive Specrtal Bisection gives excellent partitionings.

On the other hand, it is rather expensive because of the requirement to compute eigenvec-

tors.

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There exist a number of other techniques which, like spectral techniques, are also based

on the adjacency graph only. The simplest idea is one that is borrowed from the technique

of nested dissection in the context of direct sparse solution methods. Refer to Chapter 3

where level set orderings are described. An initial node is given which constitutes the level

zero. Then, the method recursively traverses the -th level ( ), which consists of the

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neighbors of all the elements that constitute level . A simple idea for partitioning the `

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graph in two traverses enough levels to visit about half of all the nodes. The visited nodes

will be assigned to one subdomain and the others will constitute the second subdomain.

The process can then be repeated recursively on each of the subdomains. A key ingredient

for this technique to be successful is to determine a good initial node from which to start £ }

the traversal. Often, a heuristic is used for this purpose. Recall that is the distance | x

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between vertices and in the graph, i.e., the length of the shortest path between and .

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If the diameter of a graph is de¬ned as

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