l21 - 6.896 Sublinear Time Algorithms Lecture 21 Lecturer...

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Unformatted text preview: 6.896 Sublinear Time Algorithms April 24, 2007 Lecture 21 Lecturer: Ronitt Rubinfeld Scribe: Ning Xie In this lecture, we are going to study property testing for clustering problems. 1 Clustering problems We will be working in d-dimensional Euclidean space R d . For any two points x and y in R d , their L 2-distance (Euclidean distance) is defined to be dist( x, y ) = p ∑ n i =1 ( x i − y i ) 2 . A Euclidean ball at center x ∈ R d of radius r is the set of points that within distance r from x , i.e., B b ( x ) = { x ∈ R d : dist( x, x ) ≤ b } . Let X be a set of n points in R d . Definition 1 X is ( k , b )-radius clusterable ( ( k, b )-r.c. for short) is X can be partitioned into k subsets such that each subset can be covered by a Euclidean ball of radius at most b . Given a set of n points in R d , the problem of deciding if these points are ( k, b )-r.c. is known to be NP-hard. Therefore it is unlikely to find any polynomial time algorithm for this problem. Can we find an efficient property testing algorithm for Clustering? 2 Testing algorithm First we define a subset X of R d of size n to be-far from ( k, b )-r.c. if we need to delete at least n points from X in order to make it ( k, b )-r.c. We have the following surprisingly efficient testing algorithm for Clustering: Testing Algorithm for Clustering 1 sample m = Θ( dk log dk ) points from X 2 if the sampled points are ( k, b )-r.c., PASS; else FAIL There are algorithms that find k balls with minimum radius that contain all m sampled points (known as Euclidean...
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This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Fall '04 term at MIT.

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l21 - 6.896 Sublinear Time Algorithms Lecture 21 Lecturer...

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