Approximation Algorithms for Geometric Proximity Problems: Preliminaries Introduction Convex Part II: Approximating Convex Bodies Approximations Canonical Form Quadtree-based Gold Standard New Approach David M. Mount Our Results Intuition Metric Spaces Department of Computer Science & Delone Sets Macbeath Institute for Advanced Computer Studies Regions Hilbert University of Maryland, College Park Geometry APM Queries Data Structure Joint with: Ahmed Abdelkader, Sunil Arya, and Guilherme da Fonseca Analysis Conclusions HMI-GCA Workshop 2018
Geometric Queries with Convex Bodies Preprocess a geometric set to answer queries efficiently Preliminaries Introduction Focus on convex bodies: closed, bounded convex sets: Convex Approximations Convex hull of a set of n points in R d Canonical Form Quadtree-based Intersection of a set of n closed halfspaces in R d (within an enclosure) Gold Standard New Approach Our Results Sample queries: Intuition Metric Spaces Membership/Containment: q ∈ P ?, Q ⊆ P ? Delone Sets Macbeath Regions Intersection: Q ∩ P � = ∅ ? Hilbert Geometry Extrema: Ray shooting, directional extrema (linear-programming queries) APM Queries Data Structure Distance: Directional width, longest parallel segment, separation distance Analysis Conclusions Assumptions Bodies reside in R d , where d is a constant. Bodies are full dimensional. Euclidean distance
Geometric Queries with Convex Bodies Gold Standard for exact queries: O ( n ) space and O (log n ) query time Preliminaries Introduction Good exact solutions exist in R 2 and R 3 , but not in higher dimensions: Convex Approximations The worst-case combinatorial complexity grows as O ( n ⌊ d 2 ⌋ ) Canonical Form Quadtree-based Gold Standard Point membership, halfspace emptiness, ray shooting: New Approach Our Results R 2 , R 3 : O ( n ) space, O (log n ) query time Intuition Metric Spaces R d : O ( n ) space, � O ( n 1 − 2 d ) query time [Matouˇ sek 92] Delone Sets Macbeath Regions Intersection detection of preprocessed convex polytopes: Hilbert Geometry R 2 : O ( n ) space, O (log n ) query time [Dobkin and Kirkpatrick 83] APM Queries Data Structure R 3 : O ( n ) space, O (log 2 n ) query time [Dobkin and Kirkpatrick 90] Analysis O ( n ) space, O (log n ) query time [Barba and Langerman 15] Conclusions R d : O (log n ) query time but space O ( N ⌊ d 2 ⌋ ) where N = total combinatorial complexity [Barba and Langerman 15]
Approximating Convex Bodies Given a convex body K , and ε > 0: Preliminaries Introduction Convex Inner ε -approximation: Any set K − ε ⊆ K within Hausdorff distance ε · diam( K ) of K Approximations Canonical Form Outer ε -approximation: Any set K + Quadtree-based ε ⊇ K within Hausdorff distance ε · diam( K ) of K Gold Standard New Approach Our Results The representation often suggests which. Let P be point set, and H a set of halfspaces Intuition Metric Spaces ε = conv( P ′ ) for some P ′ ⊆ P Delone Sets K = conv( P ): Inner approximation K − Macbeath Regions K = � ( H ): Outer approximation K + ε = � ( H ′ ), for some H ′ ⊆ H Hilbert Geometry APM Queries Data Structure Analysis Many queries are equivalent through point-hyperplane duality Conclusions Most results can be adapted to any combination inner/outer, point/halfspace
Approximate Geometric Queries Preliminaries ε -Approximate Query Introduction Convex Approximations An answer is valid if it is consistent with any ε -approximation to K Canonical Form Quadtree-based Gold Standard It is often useful to have a directionally sensitive notion of approximation New Approach Our Results Given a vector v , define width v ( K ) to be the minimum distance between two Intuition Metric Spaces hyperplanes orthogonal to v that enclose K . Delone Sets Macbeath Regions Width-sensitive (outer) ε -approximation: Any set K + ε ⊇ K such that Hilbert Geometry width v ( K + ) ≤ (1 + ε ) · width v ( K ), for all v . APM Queries Data Structure Analysis Width-Sensitive Approximation Conclusions An answer is valid if it is consistent with any width-sensitive ε -approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based Gold Standard K New Approach Our Results γ Intuition 2 O Metric Spaces Delone Sets Macbeath 1 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based Gold Standard K New Approach Our Results γ Intuition 2 O K Metric Spaces Delone Sets Macbeath 1 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based Gold Standard K New Approach E Our Results γ Intuition 2 O K Metric Spaces Delone Sets Macbeath 1 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based d · E Gold Standard K New Approach E Our Results γ Intuition 2 O K Metric Spaces Delone Sets Macbeath 1 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based d · TE Gold Standard K TK New Approach TE Our Results 1 γ Intuition 2 2d O O Metric Spaces Delone Sets Macbeath 1 1 2 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form Quadtree-based ( TK ) + Gold Standard K ε TK New Approach Our Results γ Intuition 2 O Metric Spaces Delone Sets Macbeath 1 ε 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
Preconditioning - Canonical Form γ -Canonical Form Preliminaries K is nested between two origin-centered balls of radii γ/ 2 and 1 / 2 Introduction Convex Approximations Canonical Form K + Quadtree-based ε ( TK ) + Gold Standard T − 1 K ε TK New Approach K Our Results γ Intuition 2 O Metric Spaces Delone Sets Macbeath 1 ε 2 Regions Hilbert Geometry APM Queries Data Structure Can convert to 1 d -canonical form in O ( n ) time — John’s Theorem + fast minimum Analysis enclosing/enclosed ellipsoid [Chazelle and Matouˇ sek 1996] Conclusions Since diameter ≤ 1, can use absolute error of ε Uniform approximation to TK induces a width-sensitive approximation to K
First Stab - Quadtree-based Approximation Preliminaries Introduction Query: ε -Approximate Polytope Membership ( ε -APM) Convex Approximations Canonical Form Preprocessing: Build a quadtree, subdividing each Quadtree-based node that cannot be resolved as being inside or outside Gold Standard K New Approach Stop at diameter ε Our Results Intuition Query: Find the leaf node containing q and return its Metric Spaces Delone Sets label Macbeath Regions Hilbert Geometry Performance: APM Queries Query time: O (log 1 Data Structure ε ) — Quadtree descent Analysis Storage: O (1 /ε d − 1 ) — No. of leaves ← − independent of n Conclusions
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