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 ⊆ R d • 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 � 1 � � O (1) � - Strong LTAS, running time O n + ǫ �� 1 � O (1) � - Coreset size O ǫ - 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 ∈ R d − 1 w ( u , P ) = max p ∈ P � [ u , 1] , p � − min p ∈ P � [ u , 1] , p � • Extent of a set F of ( d − 1)-variate functions at x ∈ R d − 1 e ( x , F ) = max f ∈ F f ( x ) − min f ∈ F f ( x )

Optimization Primitives • Q ⊆ P is an ǫ -approximation for P on ∆ ⊆ R d − 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 ∆ ⊆ R d − 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 ∈ R d is α -fat if there exists a translation t such that α C ⊆ 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 of an arbitrary coreset - The size and construction time are clearly preserved

Step 1: There Exists a “Fattening Transform” • [Barequet, Har-Peled 01]: Let P ⊆ R d be of size n . Can compute in O ( n ) time a box B and a vector t ∈ R d such that α B ⊆ CH ( P + t ) ⊆ B B CH (P) • Recall: α = 1 / 10? for d = 3 αB • 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 ∆ ⊆ R d − 1 if and only if T ( Q ) ǫ -approximates T ( P ) over { v | [ v , 1] = M T [ u , 1] , u ∈ ∆ } • Proof: Easy by definition - Note: can assume T ( x ) = Mx - Also, � [ u , 1] , Mp � = � M T [ u , 1] , p �

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n • Lemma [Rough approximation]: w ( x , P ) ≥ 2 α || x || max p ∈ P � x , p � = || x || · max p ∈ P � x ∀ x ∈ R d : || x || , p � ≥ || x || α � �� � projection For x d = 1: max p ∈ P � x , p � ≥ α || x || min p ∈ P � x , p � ≤ − α || x ||

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n • Lemma [Rough approximation]: w ( x , P ) ≥ 2 α || x || max p ∈ P � x , p � = || x || · max p ∈ P � x ∀ x ∈ R d : || x || , p � ≥ || x || α � �� � projection For x d = 1: max p ∈ P � x , p � ≥ α || x || min p ∈ P � x , p � ≤ − α || x || • Lemma [Hausdorff dist.]: If max p ∈ P min q ∈ Q || p − q || ≤ ǫα then Q is an ǫ -approximation for P

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n • Lemma [Rough approximation]: w ( x , P ) ≥ 2 α || x || max p ∈ P � x , p � = || x || · max p ∈ P � x ∀ x ∈ R d : || x || , p � ≥ || x || α � �� � projection For x d = 1: max p ∈ P � x , p � ≥ α || x || min p ∈ P � x , p � ≤ − α || x || • Lemma [Hausdorff dist.]: If max p ∈ P min q ∈ Q || p − q || ≤ ǫα then Q is an ǫ -approximation for P w ( x , P ) − w ( x , Q ) ≤ � x , p 1 − p 2 � − � x , q 1 − q 2 �

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n • Lemma [Rough approximation]: w ( x , P ) ≥ 2 α || x || max p ∈ P � x , p � = || x || · max p ∈ P � x ∀ x ∈ R d : || x || , p � ≥ || x || α � �� � projection For x d = 1: max p ∈ P � x , p � ≥ α || x || min p ∈ P � x , p � ≤ − α || x || • Lemma [Hausdorff dist.]: If max p ∈ P min q ∈ Q || p − q || ≤ ǫα then Q is an ǫ -approximation for P w ( x , P ) − w ( x , Q ) ≤ � x , p 1 − p 2 � − � x , q 1 − q 2 � ≤ |� x , p 1 − p 2 �| − |� x , q 1 − q 2 �|

The Case of α -fat Point Sets • From now on assume α C ⊆ CH ( P ) ⊆ C where α depends only on d , not on n • Lemma [Rough approximation]: w ( x , P ) ≥ 2 α || x || max p ∈ P � x , p � = || x || · max p ∈ P � x ∀ x ∈ R d : || x || , p � ≥ || x || α � �� � projection For x d = 1: max p ∈ P � x , p � ≥ α || x || min p ∈ P � x , p � ≤ − α || x || • Lemma [Hausdorff dist.]: If max p ∈ P min q ∈ Q || p − q || ≤ ǫα then Q is an ǫ -approximation for P w ( x , P ) − w ( x , Q ) ≤ � x , p 1 − p 2 � − � x , q 1 − q 2 � ≤ |� x , p 1 − p 2 �| − |� x , q 1 − q 2 �| ≤ |� x , ( p 1 − q 1 ) − ( p 2 − q 2 ) �| ≤ || x || · 2 αǫ

Algorithm 1: Grid ǫα • Grid of cell size √ 2 d