Approximating Extent Measures of Points Pankaj K. Agarwal Sariel - - PowerPoint PPT Presentation

Approximating Extent Measures of Points

Pankaj K. Agarwal Sariel Har-Peled Kasturi R. Varadarajan CS468, Winter 2006

Extent Measures

• Given a point set P ⊆ Rd
• Extent measures – statistics about P or its enclosing shape
• Some examples
• k-th largest distance
• Min. volume bounding box
• Min. width bounding slab
• Min. enclosing sphere
• Min. enclosing cylinder
• Min. width enclosing spherical shell
• Min. width enclosing cylindrical shell

Main Result

• Technique for ǫ-approximating (a large class of) extent

measures

• Compute a subset of the input Q ⊆ P (coreset) which
• Preserves the solution to ǫ-accuracy
• Small size (does not depend on |P|, only on ǫ)
• General properties
• Strong LTAS, running time O
• n +

1

ǫ

O(1)

• Coreset size O

1

ǫ

O(1)

• Exponents depend on d
• Simple to implement, and some improvements
• Min. enclosing spherical shell

exponent down from O(d 2) [Chan 02] to O(d)

Key Definitions

• Lead to two main approximation primitives
• Directional width of a point set P in the direction u ∈ Rd−1

w(u, P) = maxp∈P[u, 1], p − minp∈P[u, 1], p

• Extent of a set F of (d − 1)-variate functions at x ∈ Rd−1

e(x, F) = maxf ∈F f (x) − minf ∈F f (x)

Optimization Primitives

• Q ⊆ P is an ǫ-approximation for P on ∆ ⊆ Rd−1 if for all

u ∈ ∆ (1 − ǫ)w(u, P) ≤ w(u, Q) ≤ w(u, P)

• G ⊆ F is an ǫ-approximation for F on ∆ ⊆ Rd−1 if for all

x ∈ ∆ (1 − ǫ)e(x, F) ≤ w(u, G) ≤ e(x, F)

• Note: Always pick a subset of the input – coreset

Classes of Extent Measures

• Faithful
• Approximated through directional width
• “Convex” measures (bounding shapes)
• Other
• Approximated through extent
• “Concave” measures (“shells”)

Overview

(A) Strong LTASs for directional width

• Reduction to “fat” point sets
• Algorithm 1: Grid
• Algorithm 2: Polytope
• Algorithm 3: Decomposition

Strong LTASs for extent

• Linear functions (hyperplanes)
• Polynomial functions
• r-th roots of polynomials

(B) Dynamic updates Applications to specific extent measures

Reduction to “Fat” Point Sets

• A point set P ∈ Rd is α-fat if there exists a translation t such

that αC ⊆ CH(P) + t ⊆ C = [−1, 1]d

• Sufficient to consider computing coresets for α-fat point

sets

• Step 1:

Every point set can be made α-fat by applying a linear transformation, where α = α(d)

• Step 2:

Every linear transformation preserves the approximation ratio

• f an arbitrary coreset
• The size and construction time are clearly preserved

Step 1: There Exists a “Fattening Transform”

• [Barequet, Har-Peled 01]: Let P ⊆ Rd be of size n. Can

compute in O(n) time a box B and a vector t ∈ Rd such that αB ⊆ CH(P + t) ⊆ B

B

αB

CH(P)

• Recall:

α = 1/10? for d = 3

• Choose T so that

T(B) = C

Step 2: Invariance Under Linear Transforms

• Lemma:

Let T(x) = Mx + b be a non-degenerate linear transform. Q ⊆ P ǫ-approximates P over ∆ ⊆ Rd−1 if and only if T(Q) ǫ-approximates T(P) over {v | [v, 1] = MT [u, 1], u ∈ ∆}

• Proof: Easy by definition
• Note: can assume T(x) = Mx
• Also, [u, 1], Mp = MT[u, 1], p

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n
• Lemma [Rough approximation]: w(x, P) ≥ 2α||x||

∀x ∈ Rd: maxp∈Px, p = ||x|| · maxp∈P x ||x||, p

• projection

≥ ||x||α For xd = 1: max

p∈P x, p ≥ α||x||

min

p∈Px, p ≤ −α||x||

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n
• Lemma [Rough approximation]: w(x, P) ≥ 2α||x||

∀x ∈ Rd: maxp∈Px, p = ||x|| · maxp∈P x ||x||, p

• projection

≥ ||x||α For xd = 1: max

p∈P x, p ≥ α||x||

min

p∈Px, p ≤ −α||x||

• Lemma [Hausdorff dist.]: If maxp∈P minq∈Q ||p − q|| ≤ ǫα

then Q is an ǫ-approximation for P

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n
• Lemma [Rough approximation]: w(x, P) ≥ 2α||x||

∀x ∈ Rd: maxp∈Px, p = ||x|| · maxp∈P x ||x||, p

• projection

≥ ||x||α For xd = 1: max

p∈P x, p ≥ α||x||

min

p∈Px, p ≤ −α||x||

• Lemma [Hausdorff dist.]: If maxp∈P minq∈Q ||p − q|| ≤ ǫα

then Q is an ǫ-approximation for P w(x, P) − w(x, Q) ≤ x, p1 − p2 − x, q1 − q2

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n
• Lemma [Rough approximation]: w(x, P) ≥ 2α||x||

∀x ∈ Rd: maxp∈Px, p = ||x|| · maxp∈P x ||x||, p

• projection

≥ ||x||α For xd = 1: max

p∈P x, p ≥ α||x||

min

p∈Px, p ≤ −α||x||

• Lemma [Hausdorff dist.]: If maxp∈P minq∈Q ||p − q|| ≤ ǫα

then Q is an ǫ-approximation for P w(x, P) − w(x, Q) ≤ x, p1 − p2 − x, q1 − q2 ≤ |x, p1 − p2| − |x, q1 − q2|

The Case of α-fat Point Sets

• From now on assume αC ⊆ CH(P) ⊆ C where α depends
• nly on d, not on n
• Lemma [Rough approximation]: w(x, P) ≥ 2α||x||

∀x ∈ Rd: maxp∈Px, p = ||x|| · maxp∈P x ||x||, p

• projection

≥ ||x||α For xd = 1: max

p∈P x, p ≥ α||x||

min

p∈Px, p ≤ −α||x||

• Lemma [Hausdorff dist.]: If maxp∈P minq∈Q ||p − q|| ≤ ǫα

then Q is an ǫ-approximation for P w(x, P) − w(x, Q) ≤ x, p1 − p2 − x, q1 − q2 ≤ |x, p1 − p2| − |x, q1 − q2| ≤ |x, (p1 − q1) − (p2 − q2)| ≤ ||x|| · 2αǫ

Algorithm 1: Grid

• Grid of cell size

ǫα 2 √ d

Algorithm 1: Grid

• Grid of cell size

ǫα 2 √ d

• Clear all “internal” cells
• Convex hull moves by

at most one cell diameter ǫα/2

Algorithm 1: Grid

• Grid of cell size

ǫα 2 √ d

• Clear all “internal” cells
• Convex hull moves by

at most one cell diameter ǫα/2

• Eliminate duplicates in

the “boundary” cells

• Another shift of at

most ǫα/2

• ✁●✁●✁●
• ✁●✁●✁●
• ✁●✁●✁●
• ✁●✁●✁●
• ✁●✁●✁●

Algorithm 1: Grid

• Grid of cell size

ǫα 2 √ d

• Clear all “internal” cells
• Convex hull moves by

at most one cell diameter ǫα/2

• Eliminate duplicates in

the “boundary” cells

• Another shift of at

most ǫα/2

• Directional width changes at most ǫα
• Determined by convex hull

Algorithm 1: Analysis

• Coreset size O(1/(αǫ)d−1)
• One point per “boundary” grid cell
• Cell size O(1/(αǫ))
• Running time O(n + 1/(αǫ)d−1)
• O-notation hides

√ d

• α has a bad dependence on d

Algorithm 2: Polytope

• Run Algorithm 1, return (ǫ/2)-approximation P1
• Apply [Dudley 1974] to P1, lose another ǫ/2
• Sample the sphere of radius

√ d + 1

• Closest point routine [G¨

artner 1995], runs in linear time, returns all (at most d) closest points

• Return the set of closest points P2

Algorithm 2: Analysis

• Correctness
• Fact 1: Dudley polytope is also valid for P2
• Fact 2: CH(P2) ⊆ CH(P1) ⊆ Dudley
• Want (αǫ/2)-approximation of the convex hull
• Yields ǫ/2 for directional width
• Required number of samples

O((

• αǫ/2)d−1) = O((αǫ)(d−1)/2)
• This is also (roughly) the size of the result P2
• At most d per sample
• Compare to Algorithm 1: O((αǫ)d−1)

Algorithm 2: Analysis

• Find closest points for O((αǫ)(d−1)/2) samples
• The closest point routine [G¨

artner 1995]

• Linear in the number of points
• In this case, |P1| = O((αǫ)d−1)
• Total, O((αǫ)3(d−1)/2)
• Including the call to Algorithm 1, we get O(n + (αǫ)3(d−1)/2)
• Compare to Algorithm 1: O(n + (αǫ)d−1)

Algorithm 3: Arrangement

• Run Algorithm 2, get an (ǫ/2)-approximation P2
• Decompose a set of directions Rd−1 using an arrangement of

(d − 2)-flats u, v ∈ Rd−1 in the same cell ⇒

• u

||u|| − v ||v||

• ≤

αǫ 4 √ d

• Identify diametrically opposite cells

(induced partition of Pd−1, the set of unoriented directions)

• Return P3, consisting of two points per cell
• Extremal points in one arbitrary direction within the cell
• Lose another ǫ/2

angle error, extreme point error ≤ ≤

• u

||u|| − v ||v||

• · ||p − q|| ≤

αǫ 4 √ d · 2 √ d = αǫ 2

Algorithm 3: Computing the Arrangement

Algorithm 3: Computing the Arrangement

• (d − 1)-dimensional grid within each of the d facets of C
• Cell size

αǫ 4 √ d

• Results in a set of (d − 2)-flats
• Affine hull with the origin, intersect with Pd−1

Algorithm 3: Computing the Arrangement

• (d − 1)-dimensional grid within each of the d facets of C
• Cell size

αǫ 4 √ d

• Results in a set of (d − 2)-flats
• Affine hull with the origin, intersect with Pd−1
• Complexity of the arrangement
• Number of hyperplanes d(d − 1) 4

√ d αǫ

• Complexity O(1/(αǫ)d−1)

Algorithm 3: Analysis

• Coreset size |P3| = O(1/(αǫ)d−1)
• Same as the arrangement complexity
• Extremal points are computed by linear search through P2
• |P2| = O(1/(αǫ)(d−1)/2) time per cell
• O(1/(αǫ)d−1) cells
• O(1/(αǫ)3(d−1)/2) total
• Combined with Algorithm 2, O(n + 1/(αǫ)3(d−1)/2)
• Note |P3| > |P2|, a point may be extremal in multiple cells

Approximating (d − 1)-variate Functions

• Recall: ǫ-approximation in terms of extent

Approximating (d − 1)-variate Functions

• Recall: ǫ-approximation in terms of extent
• 1. Linear functions (hyperplanes), using duality
• 2. General polynomials, using linearization
• 3. r-th roots of polynomials, r ∈ Z

Point-Hyperplane Duality

• The hyperplane

h : xd = a1x1 + a2x2 + · · · + ad−1xd−1 + ad corresponds to the point h∗ = (a1, a2, . . . , ad) ∈ Rd

• Lemma:

Set of hyperplanes H = {h1, h2, . . . , hn} is ǫ-approximated by K ⊂ H over ∆ ∈ Rd−1 iff H∗ is ǫ-approximated by K ∗ over ∆

• Proof:

Directional width w(u, H∗) is the same as the extent e(u, H)

• Holds for any u ∈ Rd−1 and any set of hyperplanes H in Rd
• Important to define w(·, ·) using the xd = 1 plane
• Immediately implies ǫ-approximation algorithms for linear

functions in (d − 1)-variables

Linearization

• A set of (d − 1)-variate polynomials F = {f1, f2, . . . , fn}
• Linearization of dimension k

fi(x) = a(i)

0 + a(i) 1 φ1(x) + · · · + a(i) k φk(x)

i = 1, 2, . . . , n

• Reduces to a set of k-variate linear functions

fi(y) = a(i)

0 + a(i) 1 y1 + · · · + a(i) k yk

i = 1, 2, . . . , n with y = φ(x), φ : Rd−1 → Rk

• An ǫ-approximation for {fi(y)} over ∆ ∈ Rk implies an

ǫ-approximation for {fi(x)} over φ−1(∆ ∩ φ(Rd−1))

Linearization Example

• fi(x) is the distance of the point (x1, x2) ∈ R2 to a circle in

R2 with center (p(i), q(i)) and radius r (i) fi(x) = (r (i))2 − (x1 − p(i))2 − (x2 − q(i))2

• Can be written as

fi(x) = (r (i) − p(i) − q(i))2 + (2p(i))x1 + (2q(i))x2 − [x2

1 + x2 2]

• A linearization of dimension k = 3 with

a(i) = [(r (i) − p(i) − q(i))2, 2p(i), 2q(i), −1] φ(x) = [x1, x2, x2

1 + x2 2]

• [Agarwal, Matouˇ

sek 1994] Computing linearization of minimum dimension

Polynomials: Algorithms

• For a set F of n (d − 1)-variate polynomials admits a

linearization of dimension k, can compute

• Algorithm 1:

an ǫ-approximation of size O(1/ǫk), in time O(n + 1/ǫk)

• Algorithm 2:

an ǫ-approximation of size O(1/ǫk/2), in time O(n + 1/ǫ3k/2)

• Algorithm 3:

a set of O(1/ǫ)(d − 2)-dimensional surfaces in Rd−1, in time O(n + 1/ǫ3k/2), such that within each cell of their arrangement F is ǫ-approximated by 2 of its elements

• Complexity of the arrangement O(1/ǫd−1) [Agarwal, Sharir 00]

Cells are of complexity O(1)

• Algorithms 2 and 3:

an ǫ-approximation of size O(1/ǫmin{k/2,d−1}), in time O(n + 1/ǫ3k/2)

Roots of Polynomials

• Want ǫ-approximation for F = {(f1)1/r, (f2)1/r, . . . , (fn)1/r}

where r is integer

• Cannot linearize directly
• Special cases can be handled [Chan 02]
• If a, b, A, B ≥ 0 and [A, B] ⊆ [a, b] then

B−A ≥ (1−δ)(b−a) ⇒ B1/r −A1/r ≥ (1−ǫ)(b1/r −a1/r) for δ =

• ǫ

2(r−1)

r

• It suffices to compute O(ǫr) approximation to {f1, f2, . . . , fn}

Roots of Polynomials: Algorithms

• Set of F, |F| = n, contains r-th roots of (d − 1)-variate

non-negative polynomials that admit a linearization of of dimension k

• Algorithm 1:

ǫ-approximation in time O(1/ǫkr) and size O(1/ǫkr)

• Algorithms 2 and 3:

ǫ-approximation in time O(1/ǫ3kr/2) and size O(1/ǫr min{d−1,k/2})

• Algorithm 3:

A set of O(1/ǫr) (d − 2)-dimensional surfaces in Rd−1, in time O(1/ǫ3kr/2) , such that within each cell of their arrangement, F is ǫ-approximated by two of its elements

Classes of Extent Measures

• Faithful
• Approximated through directional width
• “Convex” measures (bounding shapes)
• Examples:

Diameter Width Radius of the smallest enclosing ball Volume of the minimum bounding box Volume enclosed by the convex hull Surface area of the convex hull

• Other
• Approximated through extent
• “Concave” measures (“shells”)
• Examples:

Minimum width spherical shell (annulus) Minimum width cylindrical shell

Dynamic Updates

• Based on a balanced tree data structure
• Both insertions and deletions
• Recompute approximations along the unique path to the root
• Update time per operation O((logk+1 n/ǫ)k + f (log n/ǫ) log n)
• Amortized O((logk n/ǫ)k + f (log n/ǫ))
• Based on the ranked subsets data structure
• Insertions only
• Partition the points into ranked subsets, based on the binary

encoding of n

• Data structure size O(log2k+1 n/ǫk)
• Coreset size O(log2k+1 n/ǫk) ,

amortized insertion time O((1/ǫ)k + f (ǫ))

• Coreset size O(1/ǫk) ,

amortized insertion time O(log2k+1 n/ǫk + f (ǫ))