\section{A Step-by-Step Example}

On the following page we illustrate the partitioning, placement, and routing
of a finite-element graph with roughly 5000 nodes on an $8 \times 8$ torus.
The graph was designed to calculate stresses in a
cracked plate, and is hundreds of times more dense at the center (at the
tip of the crack) than near the edges.  The purpose of this small example is 
to show the behavior of our partitioning method on an irregular grid 
with a large variation in density.

The first figure illustrates the original finite-element graph for the
crack problem.  Note that the graph is so dense near the center that the
figure is completely black.  To its right is the same graph after having
been partitioned into 64 blocks.  Due to density variations in the graph,
many of the partition blocks are indistinguishable.  The
third figure illustrates the 64-node 
minor derived from collapsing all the nodes in
each block of the partition to a single super node.  Finally, the last
figure shows a routing of this graph on an $8 \times 8$ torus.  Edges
extending past the boundary of the grid wrap around to the other side.

