Coresets for k-Means and k-Median Clustering and their Applications

50 Pages·2012·0.62 MB·English

Coresets for k-Means
and k-Median
Clustering and their
Applications
Sariel Har-Peled
Soham Mazumdar
UIUC
d
Compute C = fc ; : : : ; c g (cid:18) IR - centers
1 k
Minimize price: (cid:23) (P; k) = min dist(p; C)
opt
d
C(cid:18)IR ;jCj=k
X
p2P
where dist(p; C) = min kpc k
i i
Advantages: Less sensitive to noise. Theoretically
nice.
Problem
k median clustering
P d
Input: - set of n points in IR .
k - number of clusters
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.1
Advantages: Less sensitive to noise. Theoretically
nice.
Problem
k median clustering
P d
Input: - set of n points in IR .
k - number of clusters
d
Compute C = fc ; : : : ; c g (cid:18) IR - centers
1 k
Minimize price: (cid:23) (P; k) = min dist(p; C)
opt
d
C(cid:18)IR ;jCj=k
X
p2P
where dist(p; C) = min kpc k
i i
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.1
Problem
k median clustering
P d
Input: - set of n points in IR .
k - number of clusters
d
Compute C = fc ; : : : ; c g (cid:18) IR - centers
1 k
Minimize price: (cid:23) (P; k) = min dist(p; C)
opt
d
C(cid:18)IR ;jCj=k
X
p2P
where dist(p; C) = min kpc k
i i
Advantages: Less sensitive to noise. Theoretically
nice.
k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.1
[Arora et al. (1998)]
O(1=")+1
O n
(cid:16) (cid:17)
[Kolliopoulos and Rao (1999)]
O(% (cid:1) n log n log k) (Discrete) where
d(cid:0)1
% = exp [O((1 + log 1=")=") ]
Our result: O(1) O(1)
O n + %k log n
(cid:16) (cid:17)
(1 + ")-approx k-Median
Motivated by [Arora (1998)] - Approx. TSP
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.2
[Kolliopoulos and Rao (1999)]
O(% (cid:1) n log n log k) (Discrete) where
d(cid:0)1
% = exp [O((1 + log 1=")=") ]
Our result: O(1) O(1)
O n + %k log n
(cid:16) (cid:17)
(1 + ")-approx k-Median
Motivated by [Arora (1998)] - Approx. TSP
[Arora et al. (1998)]
O(1=")+1
O n
(cid:16) (cid:17)
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.2
Our result: O(1) O(1)
O n + %k log n
(cid:16) (cid:17)
(1 + ")-approx k-Median
Motivated by [Arora (1998)] - Approx. TSP
[Arora et al. (1998)]
O(1=")+1
O n
(cid:16) (cid:17)
[Kolliopoulos and Rao (1999)]
O(% (cid:1) n log n log k) (Discrete) where
d(cid:0)1
% = exp [O((1 + log 1=")=") ]
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.2
(1 + ")-approx k-Median
Motivated by [Arora (1998)] - Approx. TSP
[Arora et al. (1998)]
O(1=")+1
O n
(cid:16) (cid:17)
[Kolliopoulos and Rao (1999)]
O(% (cid:1) n log n log k) (Discrete) where
d(cid:0)1
% = exp [O((1 + log 1=")=") ]
Our result: O(1) O(1)
O n + %k log n
(cid:16) (cid:17)
k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.2
(1 + ")-approx k-Median
High Dimension
[Ba˘ doiu et al. (2002)]
(k=")O(1) O(1) O(k)
2 d n log n
k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.3
d
Compute: C = fc ; : : : ; c g (cid:18) IR - centers
1 k
2
Price: (cid:23) (P; k) = min (dist(pC))
opt
d
C(cid:18)IR ;jCj=k
X
p2P
where dist(p; C) = min kpc k
i i
Advantages: Less sensitive to noise.
Efficient heuristic: Lloyd’s method.
k-means clustering
d
Input: P - set of n points in IR .
k - number of cluster
O k k
Coresetsfor -Meansand -MedianClusteringandtheirApplications–p.4