1. Problem Definition of Clustering:
Informal goal: Given n "points" [Web pages, images, genome fragments, etc.] classify into "coherent groups" -- cluster
Assumptions:
(1) As input, given a (dis)similarity measure -- a distance d(p , q) between each point pair.
(2) Symmetric [i.e., d(p , q) = d(q , p)] (Examples: Euclidean distance, genome similarity, etc)
Same cluster ==> "nearby"
2. Max-Spacing k-Clusterings
k-clustering : the # of desired clusters is k
separated pair : Call points p & q separated if they're assigned to dierent clusters.
Spacing : The spacing of a k-clustering is min (separated p,q){ d(p , q) }. (The bigger the better)
Max-Spacing k-Clusterings problem : Given a distance measure d and k, compute the k-clustering with maximum spacing.
3. A Greedy Algorithm
-- Initially, each point in a separate cluster
-- Repeat until only k clusters:
-- Let p , q = closest pair of separated points (determines the current spacing)
-- Merge the clusters containing p & q into a single cluster.
Note: Just like Kruskal's MST algorithm, but stopped early.
4. Correctness of Greedy Clustering
-- Let C1, ... , Ck = greedy clustering with spacing S. Let C1', ... , Ck' = arbitrary other clustering.
Need to show : spacing of C1', ... , Ck' <= S
-- Case 1: Ci' are the same as the Ci (maybe after renaming) ==> has the same spacing S.
-- Case 2: Otherwise, can find a point pair p , q such that:
(A) p , q in the same greedy cluster Ci
(B) p , q in different clusters Ci'
-- Easy case: If p , q directly merged at some point in Ci, then S >= d(p , q) (Distance between merged point pairs only goes up) == > S >= spacing of C1', ... , Ck' ( since p, q are separated )
-- Tricky case: p , q "indirectly merged" through multiple direct merges. Let p, a1, ... al, q be the path of direct greedy merges connecting p & q. Since p in Ci' and q not in Ci' ==> exists consecutive pair aj , aj+1 with aj in Ci' and aj+1 not in Ci' ==> S >= d(aj , aj+1) >= Spacing of C1', ... , Ck'
