1. BIRCH: An Efficient Data Clustering Method for Very Large Databases
Tian Zhang, Raghu Ramakrishnan, Miron Livny

2. Outline of the Paper Background Clustering Feature and CF Tree
The BIRCH Clustering Algorithm
Performance Studies

3. Background Question: How to cluster large Datasets? Answer: Birch
The limited Memory
Minimize the I/O cost
Answer: Birch

4. Background (Single cluster)
Given N d-dimensional data points : {Xi}
“Centroid”
“radius”
“diameter”

5. Background (two clusters)
Given the centroids : X0 and Y0,
The centroid Euclidean distance D0:
The centroid Manhattan distance D1:

6. Background ( two clusters)
Average inter-cluster distance
D2=
Average intra-cluster distance
D3=

7. Clustering Feature CF = (N, LS, SS) N =|C| “number of data points”
LS = “linear sum of N data points”
SS = “square sum of N data points ”

8. CF Additive Theorem
Assume CF1=(N1, LS1 ,SS1), CF2 =(N2,LS2,SS2) .
Information stored in CFs is sufficient to compute:
Centroids
Measures for the compactness of clusters
Distance measure for clusters

9. CF-Tree height-balanced tree two parameters:
branching factor
B : An internal node contains at most B entries [CFi, childi]
L : A leaf node contains at most L entries [CFi]
threshold T
The diameter of all entries in a leaf node is at most T
Leaf nodes are connected via prev and next pointers

10. CF tree example

11. BIRCH Algorithm Overview

12. CF tree construction
Transform a point p into a CF-vector CFi = (1, p, p2)
Set T (threshold value, diameter or radius)
B (Branching factor) and L (number of entries in a leaf node) are determined by the value of P (page size)

13. Phase 1 Start CF tree t1 of initial T
Continue scanning data and insert into t1
Out of memory
Finish scanning data
Result?
increase T
rebuild CF tree t2 of new T from CF tree t1. if a leaf entry is a
potential outlier, write to disk. Otherwise use it.
t1 <= t2
Otherwise
Result?
Out of disk space
Re-absorb potential outliers into t1
Re-absorb potential outliers into t1

14. Insertion Algorithm Identifying the appropriate leaf
Modifying the leaf: assume the closest leaf entry, say Li,
Li can `absorb' `Ent' if T is satisfied
Add a new entry for `Ent' to the leaf
Split the leaf node
Modifying the path to the leaf:
The parent has space for this entry
Split the parent, and so on up to the root

15. Phase 2 (optional)
Scans leaf entries in the initial CF tree to rebuild a smaller CF tree and remove more outliers.

16. Phase 3: Global Clustering
Use an existing global or semi-global algorithm to cluster all the leaf entries across the boundaries of different nodes.
This way we can overcome the following anomaly:
Anomaly: Depending upon the order of data input and the degree of skew, it is also possible that two subclusters that should not be in one cluster are kept in the same node.

17. Phase 4 (optional)
Additional passes over the data set to improve the quality of the clusters.
Uses centroids of the clusters produced by phase 3 as seeds.
Redistributes data points to its closest seed.

18. Comparison of BIRCH and CLARANS

19. Summary
Compared with previous distance-based approached (e.g, K-Means and CLARANS), BIRCH is appropriate for very large datasets.
BIRCH can work with any given amount of memory.