An Interval MST Procedure

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
• Kruskals adds edges in increasing order
• Can form disconnected point segments
• All fragments eventually join

4
The MST cont.

• In/Out Algorithm (Prims)
• 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 2nd 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 pairs 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
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)