1. An Interval MST Procedure Rebecca Nugent w/ Werner Stuetzle November 16, 2004

2. 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

3. The MST cont. Several common algorithms Kruskal’s adds edges in increasing order  Can form disconnected point segments  All fragments eventually join

4. The MST cont. In/Out Algorithm (Prim’s)  Start with an “in” point  Find 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 2 nd closest distance from previous step.

5. 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

6. 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

7. Finding the Minimum Grid Search Option  May not find it  Computationally expensive

8. 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

9. 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

10. Now for the white board