Lecture 3 Randomized Algorithms
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1 Lecture 3 Randomized Algorithms Jean-Daniel Boissonnat Winter School on Computational Geometry and Topology University of Nice Sophia Antipolis January 23-27, 2017 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 1 / 39
2 Outline 1 Combinatorial complexity 2 Deterministic incremental construction of convex hulls 3 Randomized incremental algorithm 4 Fast point location 5 k-order Voronoi Diagrams Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 2 / 39
3 Combinatorial complexity The combinatorial complexity of the Voronoi diagram of n points of R d is the same as the combinatorial complexity of the intersection of n half-spaces of R d+1 The combinatorial complexity of the Delaunay triangulation of n points of R d is the same as the combinatorial complexity of the convex hull of n points of R d+1 Both complexities are the same by duality Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 3 / 39
4 Euler formula for 3-polytopes The numbers of vertices s, edges a and facets f of a polytope of R 3 satisfy Schlegel diagram s a + f = 2 s = s a = a +1 f = f +1 s = s +1 a a +1 f = f Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 4 / 39
5 Euler formula for 3-polytopes : s a + f = 2 Incidences edges-facets 2a 3f = a 3s 6 f 2s 4 with equality when all facets are triangles Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 5 / 39
6 Beyond the 3rd dimension Upper bound theorem [McMullen 1970] If H is the intersection of n half-spaces of R d nb faces of H = Θ(n d 2 ) Hyperplanes in general position any k-face is the intersection of d k hyperplanes defining H all vertices of H are incident to d edges and have distinct x d the affine hull of k < d edges incident to a vertex p contains a k-face of H Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 6 / 39
7 Proof of the upper bound theorem Bounding the number of vertices 1 d 2 edges incident to a vertex p are in h+ p : x d x d (p) or in h p p is a x d -max or x d -min vertex of at least one d -face of H 2 # vertices of H 2 # d -faces of H 2 2 A k-face is the intersection of d k hyperplanes defining H ( ) n # k-faces = = O(n d k ) d k # d 2 -faces = O(n d 2 ) Bounding the total number of faces The number of faces incident to p depends on d but not on n Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 7 / 39
8 1 Combinatorial complexity 2 Deterministic incremental construction of convex hulls 3 Randomized incremental algorithm 4 Fast point location 5 k-order Voronoi Diagrams Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 8 / 39
9 Computing the convex hull of n points of R d Adjacency graph (AG) of the facets In general position, all the facets are (d 1)-simplexes Adjacency graph (V, E) V = set of (d 1)-faces (facets) (f, f ) E iff f f share a (d 2)-face Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 9 / 39
10 Incremental algorithm pi P i : set of the i points that have been inserted first s e conv(p i ) : convex hull at step i O conv(ei) t f = [p 1,..., p d ] is a red facet iff its supporting hyperplane separates p i from conv(p i ) orient(p 1,..., p d, p i ) orient(p 1,..., p d, O) < 0 orient(p 0, p 1,..., p d ) = p 0 p 1... p d = x 01 x x d x 0d x 1d... x dd Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 10 / 39
11 Update of conv(p i ) red facet = facet whose supporting hyperplane separates o and p i+1 horizon : (d 2)-faces shared by a blue and a red facet Update conv(p i ) : 1 find the red facets s pi 2 remove them and create the new facets [p i+1, g], g horizon 3 create the new adjacencies O conv(ei) e t Complexity proportional to the number of red facets Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 11 / 39
12 Updating the adjacency graph We look at the d-simplices to be removed and at their neighbors The number of times a removed d-simplex is considered is equal to the number of its (d 2)-faces ( d + 1 d 1 ) = d(d+1) 2 Update cost = O(# created and deleted simplices ) = O(# created simplices) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 12 / 39
13 Complexity analysis update proportional to the number of red facets s pi # new facets = conv(i, d 1) = O(i d 1 2 ) fast locate : insert the points in lexicographic order and search a 1st red facet in star(p i 1 ) (which necessarily exists) O conv(ei) e t T (n, d) = O(n log n) + n i=1 = O(n log n + n d+1 2 ) i d 1 2 ) Worst-case optimal in even dimensions Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 13 / 39
14 Lower bound y = x 2 conv({p i}) = tri({x i}) pi = (xi, x 2 i ) the orientation test reduces to 3 comparisons xi orient(p i, p j, p k ) = xj xi x 2 i x 2 j x k x 2 i x 2 k xi = (x i x j)(x j x k )(x k x i) = Lower bound : Ω(n log n) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 14 / 39
15 Lower bound for the incremental algorithm No incremental algorithm can compute the convex hull of n points of R 3 in less than Ω(n 2 ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 15 / 39
16 Constructing Del(P ), P = {p 1,..., p n } R d Algorithm 1 Lift the points of P onto the paraboloid x d+1 = x 2 of R d+1 : p i ˆp i = (p i, p 2 i ) 2 Compute conv({ˆp i }) 3 Project the lower hull conv ({ˆp i }) onto R d Complexity : Θ(n log n + n d+1 2 ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 16 / 39
17 A pedestrian view : insertion of a new point p i 1. Location : find all the d-simplices that conflict with p i i.e. whose circumscribing ball contains p i 2. Update : construct the new d-simplices p i p i T p i Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 17 / 39
18 1 Combinatorial complexity 2 Deterministic incremental construction of convex hulls 3 Randomized incremental algorithm 4 Fast point location 5 k-order Voronoi Diagrams Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 18 / 39
19 Randomized incremental algorithm o : a point inside conv(p) P i : the set of the first i inserted points e pi conv(p i ) : convex hull at step i O conv(ei) Conflict graph bipartite graph {p j } {facets of conv(p i )} p j f j > i (p j not yet inserted) f op j Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 19 / 39
20 Randomized analysis Hyp. : points are inserted in random order Conflict : Notations R : random sample of size r of P F (R) = { subsets of d points of R} F 0 (R) = { elements of F (R) with 0 conflict in R} F 1 (R) = { elements of F (R) with 1 conflict in R} (i.e. conv(r)) C i (r, P) = E( F i (R) ) (expectation over all random samples R P of size r) Lemma C i (r, P) = O(r d 2 ), i = 1, 2 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 20 / 39
21 Proof of the lemma : C 1 (r, P) = C 0 (r, P) = O(r d 2 ) R = R \ {p} f F 0 (R ) if f F 1 (R) and p f (proba = 1 r ) or f F 0 (R) and R the d vertices of f (proba = r d r ) Taking the expectation, C 0 (r 1, R) = 1 r F 1(R) + r d F 0 (R) r C 0 (r 1, P) = 1 r C 1(r, P) + r d C 0 (r, P) r C 1 (r, P) = d C 0 (r, P) r (C 0 (r, P) C 0 (r 1, P)) d C 0 (r, P) = O(r d 2 ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 21 / 39
22 Randomized analysis 1 Updating the convex hull + memory space Expected number N(i) of facets created at step i N(i) = f F (P) = d i O ( i d 2 ) = O(n d 2 1 ) Proba(f F 0 (P i )) d i Expected total number of created facets = O(n d 2 ) O(n) if d = 2, 3 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 22 / 39
23 Randomized analysis2 Updating the conflict graph Cost proportional to the number of faces of conv(p i ) in conflict with p i+1 and some p j, j > i N(i, j) = expected number of faces of conv(p i ) in conflict with p i+1 and p j, j > i P + i = P i {p i+1 } {p j } : a random subset of i + 2 points of P N(i, j) = f F (P) ( Proba(f F 2(P + i + 2 i )) 2 ) 1 = 2 C2(i + 1) (i + 1)(i + 2) = O(i d 2 2 ) Expected total cost of updating the conflict graph n n n N(i, j) = (n i) O(i d 2 2 ) = O(n log n + n d 2 ) i=1 j=i+1 i=1 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 23 / 39
24 Theorem The convex hull of n points of R d can be computed in time O(n log n + n d 2 ) using O(n d 2 ) space The same bounds hold for computing the intersection of n half-spaces of R d The randomized algorithm can be derandomized [Chazelle 1992] Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 24 / 39
25 1 Combinatorial complexity 2 Deterministic incremental construction of convex hulls 3 Randomized incremental algorithm 4 Fast point location 5 k-order Voronoi Diagrams Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 25 / 39
26 The Delaunay hierarchy A location data structure Level 0 is Del(P) Each data point p in level l is introduced in level l + 1 with probability β = 1 α Location structure 1/α Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 26 / 39
27 Point location in the Delaunay Triangulation Location of point q: find the nearest neighbor of q in P n l (q): nearest neighbor of q in P l Locate q in the highest level From n l+1 (q) to n l (q): - use the pointer of n l+1 (q) to level l - walk in level l from n l+1 (q) to n l (q) The number of steps performed at level(l) : m l m l k if n l+1 (p) is the kth neighbor of q in P l Exp(m l ) n l k=1 k(1 β) k 1 β β [ β (1 β) k] = 1 β k Expected total number of steps: O(log n). Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 27 / 39
28 1 Combinatorial complexity 2 Deterministic incremental construction of convex hulls 3 Randomized incremental algorithm 4 Fast point location 5 k-order Voronoi Diagrams Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 28 / 39
29 k-order Voronoi Diagrams Let P be a set of sites. Each cell in the k-order Voronoi diagram Vor k (P ) is the locus of points in R d that have the same subset of P as k-nearest neighbors. Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 29 / 39
30 k-order Voronoi diagrams are power diagrams Let S 1, S 2,... denote the subsets of k points of P. The k-order Voronoi diagram is the minimization diagram of δ(x, S i ) : where b i is the ball 1 centered at c i = 1 k δ(x, S i ) = 1 (x p) 2 k p S i = x 2 2 p x + 1 k k p S i = π(b i, x) p S i p 2 with s i = π(o, b i ) = c 2 i r2 i = 1 k 3 and radius ri 2 = c2 i 1 k p S i p 2. p S i p 2 p S i p 2 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 30 / 39
31 Combinatorial complexity of k-order Voronoi diagrams Theorem If P be a set of n points in R d, the number of vertices and faces in all the Voronoi diagrams Vor j (P ) of orders j k is: O (k d+1 2 n d+1 ) 2 Proof uses : bijection between k-sets and cells in k-order Voronoi diagrams the sampling theorem (from randomization theory) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 31 / 39
32 k-sets and k-order Voronoi diagrams P a set of n points in R d k-sets A k-set of P is a subset P of P with size k that can be separated from P \ P by a hyperplane k-order Voronoi diagrams k points of P have a cell in Vor k (P ) iff there exists a ball that contains those points and only those each cell of Vor k (P ) corresponds to a k-set of φ(p ) h(σ) σ P Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 32 / 39
33 k-sets and k-levels in arrangements of hyperplanes For a set of points P R d, we consider the arrangement of the dual hyperplanes A(P ) h defines a k set P h separates P (below h) from P \ P (above h) h is below the k hyperplanes of P and above those of P \ P k-sets of P are in 1-1 correspondance with the cells of A(P ) of level k, i.e. with k hyperplanes of P above it. Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 33 / 39
34 Bounding the number of k-sets c k (P ) : Number of k-sets of P = Number of cells of level k in A(P ) c k (P ) = l k c l(p ) c k(p ) : Number of vertices of A(P ) with level at most k c k (n) = max P =n c k (P ) c k(n) = max P =n c k(p ) Hyp. in general position : each vertex d hyperplanes incident to 2 d cells Vertices of level k are incident to cells with level [k, k + d] Cells of level k have incident vertices with level [k d, k] c k (n) = O (c k(n)) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 34 / 39
35 Regions, conflicts and the sampling theorem O a set of n objects. F(O) set of configurations defined by O each configuration is defined by a subset of b objects each configuration is in conflict with a subset of O F j (O) set of configurations in conflict with j objects F k (O) number of configurations defined by O in conflict with at most k objects of O f 0 (r) = Exp( F 0 (R ) expected number of configurations defined and without conflict on a random r-sample of O. The sampling theorem [Clarkson & Shor 1992] For 2 k n b+1, F k(o) 4 (b + 1) b k b f 0 ( n k ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 35 / 39
36 Proof of the sampling theorem ( ) n b j ( ) n b k f 0 (r) = j F j (O) r b ( ) F k (O) n r r b ( ) n r ( ) n b k then,we prove that for r = n k r b ( ) n r 1 4(b + 1) b k b ( ) n b k r b ( ) = n r r! (n b)! (n r)! (n b k)! (r b)! n! (n r k)! (n b)! }{{}}{{} 1 (b+1) b k b 1 4 Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 36 / 39
37 Proof of the sampling theorem end (n r)! (n b k)! (n r k)! (n b)! = k j=1 n r k + j n b k + j ( n n/k k + 1 n k ) k ( ) k n r k + 1 n b k + 1 (1 1/k) k 1/4 pour (2 k), r! (n b)! (r b)! n! = b 1 l=0 b l=1 r l b n l n/k b n l=1 1/k b (1 bk n )b r + 1 b n 1 k b (b + 1) b pour (k n b + 1 ). Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 37 / 39
38 Bounding the number of k-sets c k (P ) : Number of k-sets of P = Number of cells of level k in A(P ). c k (P ) = l k c l(p ) c k(p ) : Number of vertices of A(P ) with level at most k. Objects O: n hyperplanes of R d Configurations : vertices in A(O), b = d Conflict between v and h : v h + ( Sampling th: c k(p ) 4(d + 1) d k d f n 0 k Upper bound th: f 0 ( ( n k ) = O n ) d 2 k d 2 ) c k(n) = O (k d 2 n d 2 ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 38 / 39
39 Combinatorial complexities Number of vertices of level k in an arrangement of n hyperplanes in R d Number of cells of level k in an arrangement of n hyperplanes in R d Total number of j k sets for a set of n points in R d O (k d 2 n d 2 ) Total number of faces in the Voronoi diagrams of order j k for a set of n points in R d ( O k d+1 2 n d+1 2 ) Computational Geometry and Topology Randomized Algorithms J-D. Boissonnat 39 / 39
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