A Space-Optimal Data-Stream Algorithm for Coresets in the Plane Pankaj K. Agarwal Joint work with Hai Yu

A Space-Optimal Data-Stream Algorithm for A Space-Optimal Data-Stream Algorithm for Coresets in the PlaneCoresets in the Plane

Pankaj K Agarwal

Joint work with Hai Yu

2

Extent MeasuresExtent Measures

Diameter width convex hull Simple shapes that enclose point sets

Min-radius ball min-volume box min-radius cylinder Min-width cylindrical shell min-width spherical shell

Maintaining shape descriptors for moving points Applications

Computer graphics Solid modeling Machine learning Data mining Sensor networks

3

ApproximationApproximation

ε-Kernel [A Har-Peled Varadarajan04]

This talk maintaining an ε-kernel Q of a stream S of points

a small subset that ε-approximate width of original input in all directions u

wu

Wu

computable an -kernel of size 1d2 in O(n+1εd-32) time [A etal Chan]

results in linear-time approx algorithm for many extent measures

4

ResultsResults Previous results

Our main result a space-optimal data-stream algorithm in R2

O(1ε12) space O(log 1ε) update time (amortized)

Dimension Space Update time Ref

Rd (1ε(d-1)2)logd

n1ε3(d-1)2 [A Har-Peled

Varadarajan 04]

Rd 1εd-32logd

(1ε)1ε12 [Chan 04]

R2 1ε12 log (1ε) [HershbergerSuri04]

5

Problem is easy if point set is fat

Algorithm OverviewAlgorithm Overview

6



7


Keep track of nearest neighbor of each grid point Can be implemented efficiently in O(log 1ε) time per point


8

Epochs and subepochs


o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1

Starting a new subepoch

o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)

Starting a new epoch

o xi

yj+1

20




o

xi+1

2||oxi+1||

21


Chanrsquos algorithm

o x

22



Too close to point o

o x

Keep last log(1ε) epochs alive

23




o x


24




Keep last log(1ε) subepochs alive


Too close to line ox

o x

25

Algorithm OverviewAlgorithm Overview Chanrsquos algorithm

Total space (1ε12) log2(1ε)




o x


26

New IngredientNew Ingredient

Prune points of previous epochs at the beginning of each new epoch

o

last log(1ε) epochs

27



o x


28



o x


29



o x


j-th last O(j2(2jε)12)

Total space remains O(1ε12)

O(1) O(1ε12)

30

New IngredientNew Ingredient Prune points of previous subepochs at the beginning of each new

subepoch

Algorithm for subepoch does not touch points of previous epochs

o x

j-th last O(j(2jε)12)

last log(1ε) subepochs

O(1) O(1ε12)

31

Point set is always stretched in vertical direction

Intuition for SubepochIntuition for Subepoch

2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32

Starting a SubepochStarting a Subepoch

Splitting and pruning π2

π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33

Point set is stretched in several directions

Intuition for EpochIntuition for Epoch

π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36


Overlay two angular partitions and maintain proper invariants


π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37

ExtensionsExtensions

Space-optimal approximation algorithm for maintaining width (smallest enclosing box etc) in R2

Space-optimal algorithm for maintaining (k ε)-kernels in R2

O(k ε12) space O(k log 1ε) update time

Improved algorithm for maintaining ε-kernels in Rd

O(1εd-32) space O(log 1ε) update time Similar results for (k ε)-kernels

Improved algorithms for numerous data-stream problems related to extent measures

2

Extent MeasuresExtent Measures

Diameter width convex hull Simple shapes that enclose point sets

Min-radius ball min-volume box min-radius cylinder Min-width cylindrical shell min-width spherical shell

Maintaining shape descriptors for moving points Applications

Computer graphics Solid modeling Machine learning Data mining Sensor networks

3





wu

Wu



4





Rd (1ε(d-1)2)logd


Varadarajan 04]

Rd 1εd-32logd

(1ε)1ε12 [Chan 04]


5



6



7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








3





wu

Wu



4





Rd (1ε(d-1)2)logd


Varadarajan 04]

Rd 1εd-32logd

(1ε)1ε12 [Chan 04]


5



6



7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








4





Rd (1ε(d-1)2)logd


Varadarajan 04]

Rd 1εd-32logd

(1ε)1ε12 [Chan 04]


5



6



7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








5



6



7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








6



7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








7




8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








8



o

9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








9



o xi

10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








10



o

2||oxi||

xi

an epoch

11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








11



o

yj

2||oxi||

xi

12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








12



o

yj

2||oxi||

xi

2d(yjoxi

)

13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








13



2||oxi||

o xi

a subepoch

2d(yjoxi

)

yj

14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








14



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








15



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








16



2||oxi||

o xi

2d(yjoxi

)

yj

a subepoch

17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








17



2||oxi||

o xi

yj 2d(yjoxi

)

a subepoch

18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








18



2||oxi||

yj+1


o xi

2d(yj+1oxi)

19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








19



2||oxi||

2d(yj+1oxi)


o xi

yj+1

20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








20




o

xi+1

2||oxi+1||

21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








21



o x

22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








22




o x


23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








23




o x


24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








24







o x

25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








25






o x


26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








26



o


27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








27



o x


28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








28



o x


29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








29



o x




O(1) O(1ε12)

30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








30


subepoch


o x



O(1) O(1ε12)

31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








31



2||ox||

o x

2d(yi-

jox)

yi-j

yi

2d(yiox)

π2π4

0π2j

hellip

32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








32



π4

0

π2j

hellip

π2

π4

0

π2j

hellip

π2j+1

33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








33



π2π4

0π2j

hellip

34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








34



π2π4

0π2j

hellip

35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








35



π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








36




π2

π4

0

hellipπ2j+1

π2π4

0π2j

hellip

37








37








Documents

A Space-Optimal Data-Stream Algorithm for Coresets in the Plane Pankaj K. Agarwal Joint work with Hai Yu