Geometric Approximation Using Coresets - CS.Duke - Duke University

Geometric Approximation Using Coresets Pankaj K. Agarwal

Department of Computer Science Duke University

Kinetic Geometry

moving points in






: Set of

 





 

Maintain the diameter (width, smallest enclosing disk) of . ✫ [A., Guibas, Hershberger, Veach] times








Kinetic data structure with



Diametral pair can change






events

✫ Can we maintain the approximate diameter of 



s.t.



Is there a small coreset

more efficiently?















 















?

✫ Kinetic bounding box hierarchies? Geometric Approximation Using Coresets

1

Shape Fitting

points in






: Set of

rI

rO



 

✫ Fit a cylinder through Find a cylinder 

 



) ( %*

 

+ 

# $

"

! 



&'

[A., Aronov,

,
-



Optimal solution: Sharir]



.










-approximation: 

✫ Can we compute an -approximation of in linear time?  

δ





?





Geometric Approximation Using Coresets

  /

approximates





so that

 

 

Is there is a small coreset

2

Geometry in Streaming Model

t=1

t=2

t=4

10

✫ An incoming stream of points in

t=3

✫ Maintain certain geometric/statistical measures of the input stream 

2

Diameter, smallest enclosing disk, -clustering 9

876


5*" 34

✫ Use

space and processing time

✫ Much work done on maintaining a summary of 1D data ✫ Little work on multidimensional geometric data [A., Krishnan, Mustafa, Venkatasubramanian], [Hershberger, Suri], [Bagchi, Chaudhary, Eppstein, Goodrich]



✫ How much storage and processing time (per point) needed to maintain -approx of smallest disk enclosing ? Maintain a core set! Geometric Approximation Using Coresets

3

L

-Approximation and Random Sampling

>=



;