algorithms for geometric data streams

Komplexitätstheorie und effiziente Algorithmen

Christian Sohler, TU Dortmund

Algorithms for geometric data streams

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Data streams• Massive data set arriving sequentially• Different ways of „arriving“

Examples• Network traffic• Query logs• …

Approach• Find algorithms that make a single (a few) pass(es) and process

data sequentially

Introduction

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Geometric data streams• Massive sets of geometric objects arriving sequentially• Objects are typically points• Different form of arrival:

- sequence of points- sequence of updates

Questions• Find ways to analyze the geometric structure of the input data using

small space

Introduction

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Motivation• Many computational tasks can be interpreted geometrically• Geometric features may be useful in learning and classification• Geometry plays an important role in the application

Examples• Learning • Clustering• How ‚clusterable‘ is a data set?• Road traffic prediction

Introduction

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A basic learning problem• We have two classes of objects

Introduction

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A basic learning problem• We have two classes of objects

Introduction

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A basic learning problem• We have two classes of objects• We are given examples from both classes

Introduction

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A basic learning problem• We have two classes of objects• We are given examples from both classes

Introduction

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A basic learning problem• We have two classes of objects• We are given examples from both classes• Learn from examples to which class

future objects belong

Introduction

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future objects belong• Map object‘s description to Euclidean space

Introduction

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future objects belong• Map object‘s description to Euclidean space

SVM approach • Compute maximum margin hyperplane• Classifiy points according to their side

Introduction

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SVM and SEB (smallest enclosing balls)• Dual of certain SVM formulation is SEB

[Tax, Duin, Pattern Recognition Letters, ‘99]• Geometric streaming SEB can be used as SVM heuristic

[Rai, Daume III, Venkatasubramanian, IJCAI‘09]• Also: Coresets have been used

to construct CSVMs[Tsang, Kwok, Cheung, Journal of Machine Learning Research, ’05]

Introduction

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Outline • Merge & Reduce• Embeddings into tree metrics• Estimation of distribution of local neighborhoods• Balanced partitions• Approximating properties of balanced partitions

Introduction

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Insertion-only streams• Sequence of points p ,…, p from R

Merge & Reduce

1 nd

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Definition [k-median clustering]Given a weighted set P of points in R the k-median problem is to find

a set CR of k points (centers) such that

cost(P,C) = wmin ||p-c||

is minimized, where w >0 is the weight of point p.

Merge & Reduce

d

pP cC

d

p

p

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Coreset [Har-Peled, Mazumdar, STOC’04]

A weighted point set S is a (k,)-coreset of a weighted point set P, if for every set C of k centers

| cost(P,C) – cost(S,C) | cost(P,C).

Merge & Reduce

3

3

3

33

4

4

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Observation• Union of two (k,)-coresets is a (k,)-coreset• Can compute coreset of a coreset

Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

Coreset

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

Coreset

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Merge & Reduce

… Input Stream

Coreset of Union of Coreset

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

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Merge & Reduce

… Input Stream

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Coresets by pre-clustering [Guha, Mishra, Motwani, O‘Callaghan, FOCS’00; Har-Peled, Mazumdar, STOC’04; Frahling, S., STOC‘05]

• Compute a pre-clustering S with >k centers and cost(P,S) Opt• Size exponential in d

Merge & Reduce

3

3

3

33

4

4

k

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Coresets by sampling [Chen, SICOMP’09; Feldman, Monemizadeh, S., SoCG‘07]

• Compute a random non-uniform sample• Show that sample approximates all solutions from a net• Size polynomial in d

Merge & Reduce

MM M/4

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Coresets by reduction to 1D [Har-Peled, Kushal, DCG’07, Feldman, Fiat, Sharir, FOCS‘06]

• Uses geometric arguments to solve 1D• Combine with preclusting using line centers• For k-median: Size independent of n (but exponential in d)

Merge & Reduce

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Open problems • Coresets for k-median of size independent of n and d ?

(Partial result in [Feldman, Monemizadeh, S., SoCG’07])• Coresets for k-median of size O(d/²)• Coresets for k-median of size poly(d, log n)/for

constant c=c(d)>0• Coresets for j-subspace 1-median of size poly(, d, j, log n) ?• Same questions for k-means objective function

Remark: Open questions refer to the definition of coresets from this talk.

Merge & Reduce

2-c

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Insertion/deletion model• Stream consists of Insert(p), Delete(p) operations• Points are from {1,…, }• Stream is consistent, i.e. no Delete(p), if p is not present and no

Insert(p), if p is already present in the current set

Geometric update streams

d

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Streaming algorithms via embeddings into tree metrics

Embeddings in tree metrics

p

q

r

s

t

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p

q

r

s

t tsr

p q

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p

q

r

s

t tsr

p q

p q sr t

2 i

2 i2 i

2 i

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p

q

r

s

t tsr

p q

p q

2 i-1

2 i2 i

2 i

qps

r

s tr

2 i-1 2 i-1 2 i-1

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p

q

r

s

t tsr

p q

p q

2 i2 i

2 i

qps

r

s tr

2 i-1 2 i-1 2 i-1

r s

2 i-22 i-2

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p

q

r

s

t tsr

p q

p q

2 i2 i

2 i

qps

r

s tr

2 i-1 2 i-1 2 i-1

r s

2 i-22 i-2

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Embeddings in tree metrics D(.,.)• ||p-q|| D(p,q)• E[D(p,q)] = O(log )||p-q||

[Bartal, FOCS’96;Charikar, Chekuri, Goel, Guha,Plotkin, FOCS’98]

tsr

p q

p q

2 i2 i

2 i

qps

r

s tr

2 i-1 2 i-1 2 i-1

r s

2 i-22 i-2

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Estimator for cost of Euclidean minimum spanning tree (EMST) [Indyk, STOC’04]

• Write EMST for cost of EMST• Write MST for cost of minimum spanning tree of tree metric D• E[MST ] = O(log ) EMST (linearity of expectation)• Use cost of MST of D as estimator

Streaming algorithms viaembeddings into tree metrics

D

D

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Observation [Indyk, STOC’04]

• The MST of D(.,.) is given by the tree defining the tree metric • #edges of length 2 = #non-empty cells in corresponding grid


p

q

r

s

t

tsr

p q

p q sr t

2 i2 i

2 i

i

2 i

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Euclidean minimum spanning tree1. Use O(log nested grids G(i) with side length 2

2. for each grid

3. approximate |G(i)| := #nonempty cells in G(i) using F sketch

4. return 2 |G(i)|

Theorem [Indyk, STOC’04]

The above algorithm computes a O(log )-approximation to the cost of the minimum spanning tree.


i

i

0

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Results using a similar approach [Indyk, STOC’04]

Earth mover‘s distance O(log )

Facility location O(log² )

Matching O(log )

k-Median O(1)

1+ with huge extraction time

Problem Approx. factor

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Streaming algorithms viaestimating the distribution of local neighborhoods

Distribution of neighborhoods • Grids G(i) as before

• R-neighborhood of C: cells within distance at most R from C

• m (i) is number of points in i-th cell of the R-neighborhood of C

1 2 3

4 5 6 7 8

9 10 11 12 13

14 15 16 17 18

19 20 21

C,R

A cell and its 2-neighborhood

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EMST estimator• Define Z (i) = ( m (i) > 0 )• EMST can be approximated from the Z (i)• Approx. ratio goes to 1 as R goes to

C,R C,R

C,R

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EMST estimator• K: Size of R-neighborhood• Z are functions from {1,…,K} to {0,1}• Random (nonempty) C defines distribution over neighborhoods,

i.e. over functions Z:{1,…,K} {0,1}• Can still estimate EMST from this distribution

C,R

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Algorithm• Sample a certain number of nonempty grid cells and maintain

number of points for each cell in their neighborhood• Sample gives estimation of the distribution of the Z (.)• Obtain estimation for EMST from estimated distribution

Theorem [Frahling, Indyk, S., IJCGA’07]

Let >0, d be constants.The cost of a Euclidean minimum spanning tree of a point set in R given as an update stream can be estimated with a factor of 1 using polylog() space.


C,R

d

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Open Problems• (1+)-approximation for matching and/or earth mover‘s distance• Other problems? Approach is not very well understood• General characterization of problems solvable via approximation of

the distribution of local neighborhoods


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Estimating the distribution [Frahling, S., STOC’05]

• Divide space into regions• For each region maintain #points inside• Balance „error“ among regions• Notion of error depends on problem

Example• 1-Median in 1D• Error cell width #points in cell

Streaming algorithms viabalanced partitions

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Small space?• Problem dependent • Need to show that decomposition in few regions with sufficiently

small error exists


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One approach [Frahling, S., STOC’05]

• Nested grids G(i)• For each grid maintain cells intersected by random sample

(sample sizes differ for different grids)• #sample points inside cell -> #points inside cell• Combine cells from different grids to space decomposition


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Works for• k-median• k-means• MaxTSP, MaxMatching, Maximum spanning tree, Average distance,

MaxCut

Why?• Require proof for k-median and k-means• Last 5 problems can be reduced to 1-median


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Approximating properties of balanced partitions[Lammersen, S., ESA‘08]• Previous approach may lead to many regions• Example: facility location• Can approximate properties of balanced partitions, e.g. #regions• Only gives approximation of cost of solution• More details in Christiane‘s talk

Streaming algorithms viaapproximation of balanced partitions

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Open problems• Min-sum-k-clustering• Other problems?


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(Some) Techniques in geometric streaming:• Merge & Reduce• Embeddings into tree metrics• Estimation of distribution of local neighborhoods• Balanced partitions• Approximating properties of balanced partitions

And lots of open problems to work on…

Summary

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Thank you!

algorithms for geometric data streams

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