We call a map of , any set , of subsets of the vertex

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When all the subsets are disjoint, the map is called a proper partition; otherwise we refer

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to it as an overlapping partition.

The most general way to describe a node-to-processor mapping is by setting up a

list for each processor, containing all the nodes that are mapped to that processor. Three

distinct classes of algorithms have been developed for partitioning graphs. An overview of

each of these three approaches is given next.

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The geometric approach works on the physical mesh and requires the coordinates of the

mesh points to ¬nd adequate partitionings. In the simplest case, for a 2-dimensional rec-

tangular grid, stripes in the horizontal and vertical direction can be de¬ned to get square

subregions which have roughly the same number of points. Other techniques utilize no-

tions of moment of inertia to divide the region recursively into two roughly equal-sized

subregions.

Next is a very brief description of a technique based on work by Miller, Teng, Thur-

ston, and Vavasis [150]. This technique ¬nds good separators for a mesh using projections

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into a higher space. Given a mesh in , the method starts by projecting the mesh points

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into a unit sphere centered at the origin in . Stereographic projection is used: A line

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graphic projection of is the point where this line intersects the sphere. In the next step, a y

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centerpoint of the projected points is found. A centerpoint of a discrete set is de¬ned

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as a point where every hyperplane passing through will divide approximately evenly.

Once the centerpoint is found, the points of the sphere are rotated so that the centerpoint is

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aligned with the North Pole, i.e., so that coordinates of are transformed into . ¢ t2i¢i¢ x

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The points are further transformed by dilating them so that the centerpoint becomes the ori-

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gin. Through all these transformations, the point remains a centerpoint. Therefore, if any

hyperplane is taken that passes through the centerpoint which is now the origin, it should

cut the sphere into two roughly equal-sized subsets. Any hyperplane passing through the a

origin will intersect the sphere along a large circle . Transforming this circle back into

the original space will give a desired separator. Notice that there is an in¬nity of circles to

choose from. One of the main ingredients in the above algorithm is a heuristic for ¬nding

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centerpoints in space (actually, in the algorithm). The heuristic that is used re-

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peatedly replaces randomly chosen sets of points by their centerpoint, which are easy dt

to ¬nd in this case.

There are a number of interesting results that analyze the quality of geometric graph

partitionings based on separators. With some minimal assumptions on the meshes, it is

possible to show that there exist “good” separators. In addition, the algorithm discussed

above constructs such separators. We start with two de¬nitions.

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A -ply neighborhood system in is a set of closed disks ,

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