an interval mst procedure rebecca nugent w/ werner stuetzle november 16, 2004
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An Interval MST Procedure
Rebecca Nugent w/ Werner Stuetzle
November 16, 2004
The Minimal Spanning Tree
The MST of a graph is the spanning tree with a minimal sum of edge weights
Essentially the “lowest cost” network to connect a group of vertices/data points.
Most commonly used with an edge weight of distance between two points
The MST cont.
Several common algorithms
Kruskal’s adds edges in increasing orderCan form disconnected point segmentsAll fragments eventually join
The MST cont.
In/Out Algorithm (Prim’s)Start with an “in” pointFind the closest “out” point. Connect the two.Now find the closest “out” point to either of the
two “in” points. Connect.Etc.Need only remember the 2nd closest distance
from previous step.
New Edge Weight
Are interested in using the MST to represent the underlying “shape” of the density of the data
Use the minimum of the density between two points as the pair’s edge weight
The MST structure should indicate the modality of the data
Points in high density areas/peaks should be “close”
Points separated by a “valley” should be “far”
If we assign the min density to a pair, a low density point in a tail will cause ties in a large number of edges – these ties are broken by Eucl. distance
Finding the Minimum
Grid Search Option
May not find itComputationally expensive
Finding the Minimum
Only need to have ordering of edge weights to find MST
(Note that any monotonic transformation of the edge weights preserves the MST structure)
Can instead find an interval bounding the minimum
Finding the Minimum
Once the intervals have been found, some may overlap.
Refine the intervals until apparent which edge to add.
May not need to refine until all intervals are non-overlapping – can be selective in choosing edges
Now for the white board
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