Approximate Centerpoints with Proofs

Approximate Centerpoints with Proofs

Don Sheehy and Gary MillerCarnegie Mellon University

Given a set S ! Rd, a centerpoint is a pointp " Rd such that every closed halfspacecontaining p contains at least n

d+1points of S.

Centerpoints


d+1points of S.

Centerpoints


d+1points of S.

Centerpoints


d+1points of S.

Centerpoints


d+1points of S.

Centerpoints


d+1points of S.

Centerpoints


d+1points of S.

CenterpointsApproximate


d+1points of S.

Centerpoints!-center

Approximate


d+1points of S.

Centerpoints!-center

!n

Approximate

Two Proofs of the Centerpoint Theorem

Helly’s TheoremGiven some convex sets in Rd such that every d + 1 sets

have common intersection, then the whole collection of setshas a common intersection.





H ! halfspaces containing more than nd

d+1points




d+1points

h ! H




d+1points

h ! H

conv(h ! S)




d+1points

h ! H

conv(h ! S)

!

h!H

h

Let S be a set of at least (d + 1)(r ! 1) + 1 points in Rd.There exists a partition of S into r subsets X1, . . . , Xr

such that!r

i=1conv(Xi) "= #

Tverberg’s Theorem


such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #



such that!r

i=1conv(Xi) "= #


Choose r = n/(d+1)It’s a centerpoint!

Not all Centerpoints are Tverberg Points


•It is coNP-Complete to test if a point is a centerpoint.



•It is NP-Complete to test if a point is a Tverberg point (even for r=2).



•It is NP-Complete to test if a point is a Tverberg point (even for r=2).

•The hardness of finding a centerpoint is an open question.

Related Work

Related Work

Low Dimensions

Related Work

O(n)-time centerpoints in R2 [Jadhav and Mukhopadhyay, 1994]

Low Dimensions

Related Work


O(n2polylog(n)) for R3 [Cole, Sharir, Yap, 1987],[Naor and Sharir, 1990]

Low Dimensions

Related Work



Low Dimensions

General Dimension

Related Work



O(nd!1) randomized time for Tukey medians, [Chan, 2004]

Low Dimensions

General Dimension

Related Work



O(nd!1) randomized time for Tukey medians, [Chan, 2004]

O((log n)log d) time for O(n/d2)-centers. [Clarkson, Eppstein,Miller, Sturtivant, Teng, 1993]

Low Dimensions

General Dimension

Given n points in Rd, we can computea n

2(d+1)2 -center in time nO(log d).

Our Result

The output includes a poly-timecheckable proof of the depth.

Radon’s TheoremLet S ! Rd with |S| " d + 2, then there exists a partitonS = (S1, S2) such that conv(S1) # conv(S2) $= %.









Radon’s Theorem

d+2!

i=1

cipi = 0

d+2!

i=1

ci = 0

Let S ! Rd with |S| " d + 2, then there exists a partitonS = (S1, S2) such that conv(S1) # conv(S2) $= %.

Radon’s Theorem

d+2!

i=1

cipi = 0

d+2!

i=1

ci = 0

I+

= {i : ci > 0}

I!

= {i : ci < 0}


Radon’s Theorem

d+2!

i=1

cipi = 0

d+2!

i=1

ci = 0

I+

= {i : ci > 0}

I!

= {i : ci < 0}

!

i!I+

cipi =!

i!I!

(!ci)pi

!

i!I+

ci =!

i!I!

(!ci)


Radon’s Theorem

d+2!

i=1

cipi = 0

d+2!

i=1

ci = 0

I+

= {i : ci > 0}

I!

= {i : ci < 0}

!

i!I+

cipi =!

i!I!

(!ci)pi

!

i!I+

ci =!

i!I!

(!ci)

x =

!

i!I+

"

ci#

j!I+ cj

$

pi =

!

i!I!

"

!ci#

j!I!!cj

$

pi


Approximating Center Points with Iterated Radon Points[ Clarkson, Eppstein, Miller, Sturtivant, Teng, 1993 ]


1. Randomly sample points into sets of d+2.2. Compute the Radon point for each set. 3. Partition the points in to sets of size d+2 and replace each set by its Radon point.4. Continue until only one point remains.5. Return that point.


1. Randomly sample points into sets of d+2.2. Compute the Radon point for each set. 3. Partition the points in to sets of size d+2 and replace each set by its Radon point.4. Continue until only one point remains.5. Return that point.

O!

n

d2

"

-center with high probability.

The Iterated-Tverberg Algorithm

This almost works.

The Iterated-Tverberg Algorithm

The Iterated-Tverberg AlgorithmBucket Bi contains !point, proof" pairs of depth 2i


Start with all the points in B0



While |Bi| ! d + 2 for any i,

Combine d + 2 points in Bi to form a !point, proof" of depth 2i+1.

Prune out the extra points and put them back in B0

Put the resulting !point, proof" in Bi+1



While |Bi| ! d + 2 for any i,

Combine d + 2 points in Bi to form a !point, proof" of depth 2i+1.

Prune out the extra points and put them back in B0

Put the resulting !point, proof" in Bi+1

Return any !point, proof" from Bh where h is the maxsuch that Bh is non-empty.

CorrectnessOutput depth >

n

2(d+1)2

Correctness

At termination, every bucket has size at most d + 1.

Output depth >n

2(d+1)2

Correctness


A proof of depth 2i contains at most 2i(d + 1) points of S.

Output depth >n

2(d+1)2

Correctness


Let h be the max such that Bh is non-empty.


Output depth >n

2(d+1)2

Correctness




Output depth >n

2(d+1)2

(Output depth = 2h)

Correctness


h!

i=0

2i(d + 1)2 ! n



Output depth >n

2(d+1)2

(Output depth = 2h)

Correctness


h!

i=0

2i(d + 1)2 ! n

2h+1 ! 1 "n

(d + 1)2



Output depth >n

2(d+1)2

(Output depth = 2h)

Correctness


h!

i=0

2i(d + 1)2 ! n

2h+1 ! 1 "n

(d + 1)2

2h>

n

2(d + 1)2



Output depth >n

2(d+1)2

(Output depth = 2h)

Timing AnalysisRunning time = n

O(log d)

Timing Analysis

Let tk be the time to compute a point of depth 2k.

Running time = nO(log d)

Timing Analysis


tk ! (d + 2)tk!1 + R + 2kP


Timing Analysis


tk ! (d + 2)tk!1 + R + 2kP

tk ! (d + 2)k + (n poly(d))


Timing Analysis


tk ! (d + 2)tk!1 + R + 2kP

tk ! (d + 2)k + (n poly(d))

tlog n

2(d+1)2= O(n(d + 2)O(log n)poly(d))


Timing Analysis


tk ! (d + 2)tk!1 + R + 2kP

tk ! (d + 2)k + (n poly(d))

tlog n




Timing Analysis


tk ! (d + 2)tk!1 + R + 2kP

tk ! (d + 2)k + (n poly(d))

tlog n



More precisely, O

!

n1+log(d+2)

(d + 1)2 log(d+2)!1

"


Also in the paper...


•Using Tverberg r-partitions instead of Radon partitions.


•Using Tverberg r-partitions instead of Radon partitions.

•Solving larger subproblems in the Iterated-Radon algorithm.

Thank you.

Technology

Approximate Centerpoints with Proofs