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##

Multiresolutional Cluster Display

We define a cut **S** across a tree **T** as a horizontal boundary
that divides **T** into a top half and a bottom half and satisfies the
following criteria: for each path **R** from the root to a leaf, **S**
intersects **R** at exactly one point.
Clearly **S** defines a partition of the dataset **E**. We may
then vary the level-of-detail (LOD) in our data display by changing the
control parameters that control the location of **S**.

Any variable that varies **S** is a candidate for the LOD control
parameter. For instance, the tree depth is one conceivable discrete control
parameter. However, it is a poor choice in some cases because the number
of nodes increase dramatically with depth. This manifests as abrupt screen
changes as the LOD switches values at higher depths of the tree.

We desire a continuous LOD control parameter that provides smooth transitions
on our data display. We define:

We then choose
as the LOD control parameter. Define *S*(*w*) as the collection
of clusters whose size *v*_{i} is less than or equal to *w*
but whose parent's size is greater than *w*. Then *S*(*w*)
is a partition of **E** that satisfies our criteria for a continuous
level-of-detail control parameter. Formally, we define *S*(*w*)
as:

Note that *S*(*v*_{max})
is a single partition comprising the entire **E**, while *S*(*v*_{min})
is a partition consisting of all the leaf nodes of **T**.

The following algorithm finds the elements of *S*(*w*) in
a recursive top-down manner:

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