Convex hulls of spheres and convex hulls of convex polytopes lying - PowerPoint PPT Presentation

Introduction Parallel polytopes Convex hull of spheres Summary, extensions & open problems Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes Menelaos I. Karavelas joint work with Eleni Tzanaki University of Crete & FO.R.T.H. OrbiCG/ Workshop on Computational Geometry INRIA Sophia-Antipolis M´ editerran´ ee, December 9, 2010 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 1 / 42

Introduction Parallel polytopes Convex hull of spheres Summary, extensions & open problems Introduction 1 Parallel polytopes 2 Convex hull of spheres 3 Summary, extensions & open problems 4 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 2 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Problem setup We are given a set Σ of n spheres in E d , such that n i spheres have radius ρ i , where 1 ≤ i ≤ m and 1 ≤ m ≤ n . The radii are pairwise distinct, i.e., ρ i � = ρ j for i � = j . The dimension d is considered fixed. Problem What is the worst-case combinatorial complexity of the convex hull CH d (Σ) of Σ , when m is fixed? Was posed as an open problem by [Boissonnat & K. 2003] The problem is interesting only for odd dimensions. Throughout the talk d ≥ 3 , and d odd. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 3 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Previous/Related work – Worst-case complexities For a point set P in E d the worst-case complexity of CH d ( P ) is Θ( n ⌊ d 2 ⌋ ) . Same for spheres with same radius. [Aurenhammer 1987]: The worst-case complexity of CH d (Σ) is O ( n ⌈ d 2 ⌉ ) . Reduction to power diagram in E d +1 . [Boissonnat et al. 1996]: Showed that CH 3 (Σ) = Ω( n 2 ) . [Boissonnat & K. 2003]: Showed that CH d (Σ) = Ω( n ⌈ d 2 ⌉ ) for all d ≥ 3 . obius diagrams in E d − 1 with Correspondence between M¨ additively weighted Voronoi cells in E d and convex hulls of spheres in E d (via inversions). obius diagrams in E d − 1 is Θ( n ⌈ d 2 ⌉ ) . Worst-case bound for M¨ OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 4 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Previous/Related work – Worst-case optimal algorithms [Chazelle 1993]: The convex hull of n points in E d , d ≥ 2 , can 2 ⌋ + n log n ) time. be computed in worst-case optimal O ( n ⌊ d Worst-case optimal algorithms already existed for d = 2 , 3 . 2 ⌉ + n log n ) time [Boissonnat et al. 1996]: Presented a O ( n ⌈ d algorithm: worst-case optimal only for even dimensions (at that point). Lifting map from spheres in E d to points in E d +1 , using the radius as the last coordinate. [Boissonnat & K. 2003]: Due to the lower bound of Ω( n ⌈ d 2 ⌉ ) for CH d (Σ) , the algorithm in [Boissonnat et al. 1996] is actually worst-case optimal for all d ≥ 2 . OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 5 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Previous/Related work – Output-sensitive algorithms Many output-sensitive algorithms for the convex hull of points. We mention those related to this presentation: [Seidel 1986]: Uses the notion of polytopal shellings to compute CH d ( P ) in O ( n 2 + f log n ) time. sek & Schwarzkopf 1992]: Improved the n 2 part of [Matouˇ 2 ⌋ +1)+ ǫ + f log n ) . Seidel’s algorithm to get O ( n 2 − 2 / ( ⌊ d [Chan, Snoeyink & Yap 1997]: Works in 4D and has running time O (( n + f ) log 2 f ) . Many more algorithms for 2D and 3D, some of which are optimal (in the output-sensitive sense). Very few output-sensitive algorithms for spheres: [Nielsen & Yvinec 1998]: Computes CH 2 (Σ) in optimal O ( n log f ) time. [Boissonnat, C´ er´ ezo & Duquesne 1992]: Gift-wrapping algorithm for computing CH 3 (Σ) ; runs in O ( nf ) time. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 6 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Our results – Some definitions We have a set of spheres Σ consisting of n spheres, such that n i ≤ n spheres have radius ρ i , where 1 ≤ i ≤ m and m ≥ 2 and m is fixed. Definition We say that ρ λ dominates Σ if n λ = Θ( n ) . Definition We say that Σ is uniquely dominated if, for some λ , n λ = Θ( n ) , and n i = o ( n ) for all i � = λ . Definition We say that Σ is strongly dominated if, for some λ , n λ = Θ( n ) , and n i = O (1)) for all i � = λ . OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 7 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Our results – Qualitative point-of-view We use the term generic worst-case complexity to refer to the worst-case complexity of CH d (Σ) where there is no restriction on the number of distinct radii in Σ . Result If Σ is dominated by at least two radii, the worst-case complexity of CH d (Σ) matches the generic worst-case complexity. Result If Σ is uniquely dominated, the worst-case complexity of CH d (Σ) is asymptotically smaller than the generic worst-case complexity. Result If Σ is strongly dominated, the worst-case complexity of CH d (Σ) matches the worst-case complexity of convex hulls of points. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 8 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Our results – Quantitative point-of-view Theorem The worst-case complexity of CH d (Σ) , for m fixed, is ⌊ d 2 ⌋ Θ( � ) . 1 ≤ i � = j ≤ m n i n j Result If Σ is dominated by at least two radii, the complexity of CH d (Σ) is Θ( n ⌈ d 2 ⌉ ) . Result If Σ is uniquely dominated, the complexity of CH d (Σ) is o ( n ⌈ d 2 ⌉ ) . Result If Σ is strongly dominated, the complexity of CH d (Σ) is Θ( n ⌊ d 2 ⌋ ) . OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 9 / 42

Introduction Problem setup Parallel polytopes Previous work & problem history Convex hull of spheres Our results Summary, extensions & open problems Our results – Methodology For the upper bound we reduce the sphere convex hull problem to the problem of computing the complexity of the convex hull of m d -polytopes lying on m parallel hyperplanes of E d +1 . Theorem Let P be a set of m d -polytopes {P 1 , P 2 , . . . , P m } lying on m parallel hyperplanes of E d +1 . The worst-case complexity of CH d +1 ( P ) is ⌊ d 2 ⌋ O ( � ) , where n i = f 0 ( P i ) , 1 ≤ i ≤ m . 1 ≤ i � = j ≤ m n i n j For the lower bound we first construct a set Σ of Θ( n 1 + n 2 ) spheres ⌊ d ⌊ d in E d such that the complexity of CH d (Σ) is Ω( n 1 n 2 ⌋ 2 ⌋ + n 2 n ) . 2 1 Then we generalize this construction for m ≥ 3 . This construction also gives a matching lower bound for the parallel polytope convex hull problem. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 10 / 42

Introduction Definitions & preliminaries Parallel polytopes Upper bound on number of faces cut by a hyperplane Convex hull of spheres Inductive proof for upper bound Summary, extensions & open problems Definitions Polytope P : the convex hull of a set of points P Face of P : intersection of P with at least one supporting hyperplane k -face: k -dimensional face trivial face: the unique face of dimension d ( P ) proper faces: faces of dimension at most d − 1 d -polytope: a polytope whose trivial face is d -dimensional vertices: 0 -faces; edges: 1 -faces facets: ( d − 1) -faces; ridges: ( d − 2) -faces; simplicial polytope: all proper faces are simplices f k ( P ) : number of k -faces of P f -vector: ( f − 1 ( P ) , f 0 ( P ) , . . . , f d − 1 ( P )) f − 1 ( P ) = 1 (empty set) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 11 / 42

Introduction Definitions & preliminaries Parallel polytopes Upper bound on number of faces cut by a hyperplane Convex hull of spheres Inductive proof for upper bound Summary, extensions & open problems Definitions (cont.) For simplicial d -polytopes we can define the h -vector: ( h 0 ( P ) , h 1 ( P ) , . . . , h d ( P )) , where k � d − i � � ( − 1) k − i h k ( P ) = f i − 1 ( P ) . d − k i =0 h k ( P ) : number of facets in a shelling of P whose restriction has size k . The elements of the f -vector determine the h -vector and vice versa. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 12 / 42