Geometric Separators and the Parabolic Lift

Transcription

1 CCCG 13, Waterloo, Ontario, August 8 1, 13 Geometric Separators and the Parabolic Lit Donald R. Sheehy Abstract A geometric separator or a set U o n geometric objects (usually balls) is a small (sublinear in n) subset whose removal disconnects the intersection graph o U into roughly equal sized parts. These separators provide a natural way to do divide and conquer in geometric settings. A particularly nice geometric separator algorithm originally introduced by Miller and Thurston has three steps: compute a centerpoint in a space o one dimension higher than the input, compute a conormal transormation that centers the centerpoint, and inally, use the computed transormation to sample a sphere in the original space. The output separator is the subset o S intersecting this sphere. It is both simple and elegant. We show that a change o perspective (literally) can make this algorithm even simpler by eliminating the entire middle step. By computing the centerpoint o the points lited onto a paraboloid rather than using the stereographic map as in the original method, one can sample the desired sphere directly, without computing the conormal transormation. 1 Geometric Separators n d+ A spherical geometric separator o a collection o n balls in R d n is a sphere S that has at least d+ balls centered inside, at least centered outside, and intersects at most O(n 1 1 d ) o them (see Section or a ormal deinition). The existence o such separators in two and three dimensions was established by Miller and Thurston, though their method was quickly adapted to higher dimensions. Across a series o papers, Miller, Thurston, Teng, and Vavasis laid out the theory o geometric separators and their applications to scientiic computing [14, 15, 13, 1, 1. This line o work is a treasure trove or computational geometers as it hinges on a novel trick that combines projective and combinatorial geometry to solve an important algorithmic problem, solving linear systems arising in inite element analysis. More generally, geometric separators give a natural way to do divide and conquer or geometric problems. They have been applied to various nearest neighbor search problems [11 as well as to mesh compression [1. Other variations o geometric separators have been used or the Traveling Salesman and Minimum Steiner Tree problems in geometric settings [18, or or packing and piercing problems [. The Miller-Thurston algorithm or computing a geometric separator maps the n points (the centers o the balls) to a unit d-sphere in R d+1 via a stereographic map. It then computes a centerpoint, which is a geometric generalization o a median (a ormal deinition is given in Section ). There exists a conormal transormation o the points in R d+1 so that this centerpoint will lie exactly at the origin. To output a separator, one samples a random unit vector in R d+1. The hyperplane through the origin normal to this vector intersects the unit d-sphere at a (d 1)-sphere. The output is just the stereographic projection o this (d 1)-sphere back to R d. With high probability, such a sphere will be a geometric separator. The one aspect o this algorithm that was let to the reader, was the linear algebra required to compute the desired conormal transormation. In the original paper, it was simply asserted that it exists. Later papers explained that it can be computed via Householder transormations and cited a textbook on matrix computations. In this paper, we show that this phase o the algorithm is entirely unnecessary. By working initially with the parabolic liting p [ p p, rather than the stereographic map, the desired sphere can be sampled directly. Related work In addition to the previously mentioned work on sphere separators and their applications by various combinations o Miller, Thurston, Teng, and Vavasis, the generalization to other types o contact graphs and other shapes o separators (in particular hypercubes) was perormed by Smith and Wormald [18. This method was reined slightly by Chan to apply separators to geometric hitting set problems [. Eppstein et al. gave a linear-time, deterministic algorithm or inding geometric separators based on core sets [5. Har-Peled gave a simple proo o the existence o geometric separators or interior-disjoint disks in the plane (that easily extends to higher dimensions) [6. Department o Computer Science and Engineering, University o Connecticut, don.r.sheehy@gmail.com

2 5 th Canadian Conerence on Computational Geometry, 13 Deinitions Points We treat points in Euclidean space as column vectors. As the algebra will be in both R d as well as R d+1, we adopt the convention that a boldace vector p is a vector o R d and an overline such as in p indicates a vector in R d+1. In particular, we write to indicate the zero vector in R d. Scalars are italic and when it is useul, we add a subscript d + 1 as in p d+1 to indicate a scalar that is the (d + 1)st coordinate o a vector in R d+1. So, or example, we will write p = [ p p d+1 to indicate that p R d+1 with its irst d coordinates matching those o p and last coordinate equal to p d+1. Whenever we speak o R d as a subspace o R d+1, it is always assumed that we mean the hyperplane {p p d+1 = } in R d+1. Projections Given a point = [ d+1, we deine Π 1 to be the stereographic map rom R d to the sphere centered at with radius d+1. That is, Π 1 (p) is the intersection o the sphere with the line through [ p and the north pole, [ d+1 (other than the pole itsel). Similarly, we deine Π to be the parabolic liting map rom R d to R d+1 that lits p to p = [ p p d+1 so that p lies on the paraboloid with ocal point and directrix {p d+1 = d+1 }. The reason or the notation comes rom the act that the parabola is the limiting case o an ellipse ormed by moving one ocal point to ininity. We will consider more general stereographic projections in Section 4. In all cases, these maps are invertible (except at the north pole). We abuse notation slightly and let Π 1 (S) denote the set {Π 1(p) p S} and similarly or Π. In particular, Π 1 (Rd ) denotes the sphere centered at with radius d+1 and Π (R d ) denotes the paraboloid with ocal point and directrix {p d+1 = d+1 }. Centerpoints Given n points in R d, a centerpoint is a point c R d such that any closed halspace containing more than nd d+1 points also contains c. The existence o centerpoints ollows rom Helly s Theorem and the pigeonhole principle. Let Centerpoint(P ) denote the set o all centerpoints o the set P. It is not known how to ind a centerpoint deterministically in polynomial time or point sets in R d. However, there is an eicient randomized algorithm [3 known or at least twenty years and some recent work on deterministic approximation algorithms [9, 16. Sphere Separators A graph separator is a subset o vertices in a graph whose removal disconnects the graph. Usually, the goal is to ind a small separator that separates the graph into roughly equal sized pieces. The most amous example o graphs admitting small separators is the Planar Separator Theorem o Lipton and Tarjan [8, which states that a separator o size O( n) is always possible or planar graphs that has at least n 3 vertices in each o the resulting components. The early work on separators was directed at solving linear systems by generalized nested dissection, a method or ordering pivots in Cholesky decomposition o sparse, symmetric, positive deinite matrices [7. O particular interest were those linear systems arising in the inite element method. These systems relected the underlying geometric structure o the problem domain and thus it was natural to look to the geometry to ind small separators [1. In particular, it oten suiced to consider various deinitions o intersection graphs o systems o balls. Let B = {B 1,..., B n } be a collection o interior disjoint balls in R d. Let S be a sphere in R d and let B I (S), B E (S), and B O (S) be the subsets o B that are interior to, exterior to, and intersecting S respectively. The ollowing theorem is a combination o the main existence result or geometric separators with the algorithm used to prove existence. We state it this way, to make clear that the geometric challenge or computing a separator this way lies in inding a stereographic projection to a sphere that has a centerpoint at the center o the sphere. Theorem 1 (Sphere Separator Thm.[14, 11, 1) Let B be a collection o n interior disjoint balls with centers P. Let v R d+1 be chosen uniormly rom the unit d-sphere. I is a point in R d+1 such that is a centerpoint o Π 1 (P ) and H = {p v (p ) = } is the hyperplane through normal to v, then the sphere has the property that S = (Π 1 ) 1 (Π 1 (Rd ) H) B O (S) = O(n 1 1/d ), and B I (S), B E (S) with probability at least 1. (d + 1)n d + For ease o exposition, we only give this simplest version o the theorem. However, it has also been proven or k-ply neighborhood systems where the balls are permitted to have up to k wise interior intersections [11 as well as or α-overlap graphs where two balls are considered neighbors i or both balls increasing the radius o one by a actor o α causes them to intersect [1. In these cases, the bounds depend on k and α respectively, but the algorithm is the same and so the results o this paper apply to these versions o the theorem as well. The version stated above, when combined with the Koebe-Andreev-Thurston embedding theorem or planar graphs is strong enough to prove the Planar Separator Theorem.

3 CCCG 13, Waterloo, Ontario, August 8 1, 13 3 The Algorithm When Archimedes quipped that he could move the earth i given a suiciently long lever, he transerred the majority o the work to the lever builders o his day. We will do likewise and assume that we are given a long lever in the orm o an algorithm to eiciently compute a centerpoint o n points in R d+1. Given such an algorithm, the heavy liting will be quite easy, and as with Archimedes, that is entirely the point. Let P = {p 1,..., p n } R d be a set o points. We irst present the Miller-Thurston algorithm or computing a separator and then present the new simpliied version. 3.1 The Miller-Thurston Algorithm Let Π be the stereographic map rom R d to the unit d-sphere centered at the origin in R d+1. First, compute c Centerpoint(Π(P )). Find an orthogonal transormation Q such that Q(c) = [ θ or some θ R. 1 θ Let D = 1+θ I, where I is the identity on Rd. Choose a random unit vector v R d+1 and let S be the d-sphere ormed by intersecting the hyperplane {p v p = } with the unit d-sphere centered at the origin. Output S = Π 1 (Q 1 Π(DΠ 1 (S ))). 3. A Simpler Algorithm Using the Parabolic Lit First, compute c = [ c [ p1 [ pn c d+1 Centerpoint( p 1,..., p n ). Next, choose a random unit vector v = [ v v d+1 R d+1. Let cd+1 c r =. v d+1 Output the sphere S with center (c rv) and radius r. In the improbable case that v d+1 =, the output is just the hyperplane {p v (p c) = }. 3.3 Correctness o the Algorithm The remainder o this section will prove that the algorithm works. According to the results o Miller et al. [1, all we need is a stereographic projection o R d to a d-sphere such that the projected points have a centerpoint at the center o the sphere. The trick presented here is that we will instead show how to ind a parabolic liting that has a centerpoint at the ocal point o the paraboloid. Then, we show that i we have a point such that Π (P ) has a centerpoint at, then Π 1(P ) also has a centerpoint at, thus giving the desired map. For any vector v R d+1 and any p R d, v (Π 1 (p) ) = i v (Π (p) ) =. That is, or sampling spheres, there is no dierence between using the stereographic map or the parabolic liting. In act, we prove a much more general statement about the equivalence o various stereographic projections in Theorem 3. Theorem Let B be a collection o n interior disjoint balls with centers P = {p 1,..., p n } R d and let S be the sphere output by the algorithm in Section 3.. Then, B O (S) = O(n 1 1/d ), and B I (S), B E (S) with probability at least 1. (d + 1)n d + Proo. Using Theorem 1, it will suice to show there exists such that S = (Π 1 ) 1 (Π 1 (Rd ) H) where H = {p v (p ) = } and Centerpoint(Π 1 (P )). The equivalence o dierent stereographic projections proven in Theorem 3 implies that it will suice to prove the same acts replacing the stereographic map Π 1 with the parabolic lit Π. We will show that the ocal point that makes this true is = [ 1 c cd+1 c, where c = [ c c d+1 is the centerpoint o Π (P ) computed in the irst step. First, we show that S = (Π ) 1 (Π (R d ) H). We will assume without loss o generality that v d+1 > since [ v [ v d+1 and v v d+1 are sampled with equal probability and both yield the same sphere. We need only check that S is the orthogonal projection o Π (R d ) H to R d. The equation or Π (R d ) is p c p d+1 = p c =, c d+1 c rv d+1 and the equation or H can be rewritten as p d+1 = v (c p) + 1 cd+1 c v d+1 = v (c p) + 1 v d+1 rv d+1. So, or a point p = [ p p d+1 in the intersection, p c = v (c p) + 1 rv d+1 v d+1 rv d+1.

4 5 th Canadian Conerence on Computational Geometry, 13 Multiplying by and completing the square twice gives p (c rv) = r v d+1 + rv = r, which is precisely the equation or S. Now, we show that the ocal point is a centerpoint o Π (P ). There are several dierent ways to show this, but one nice approach is to observe that or any hyperplane normal to v passing through there is a corresponding hyperplane H normal to [ c rv 1 passing through c. The hyperplane H intersects the paraboloid Φ = { [ p p d+1 pd+1 = p } in an ellipse that projects orthogonally to the very same sphere o radius r centered at c rv. It will suice to show or any p = [ p p H Φ, that p (c rv) = r. By the deinition o H, (c rv) p 1 p = (c rv) c 1 c d+1. Multiplying by and completing the square yields It is tangent to [ R d at the origin, has major radius R and all minor radii all equal to R 1. The stereographic map Π R through [ R, the north pole o this ellipse, rom R d to E R is well-deined as is the inverse map (except at [ R ). As R goes to ininity, E R converges to the paraboloid p d+1 = p /4. This is perhaps easier to see when one observes that E R is the set o points equidistant rom and the sphere centered at [ R 1 with radius R. As R goes to ininity, this sphere becomes a plane and the paraboloid is the set o points equidistant rom a point and a plane. See Figure 1. I we intersect the ellipsoid E R with a hyperplane through one i its ocal points and then stereographically project that intersection to R d rom the pole o the ellipse, then the result is a sphere. The ollowing theorem says that or a ixed hyperplane, it doesn t matter which ellipsoid E R we started with, we always get the same sphere as illustrated in Figure. p (c rv) = c rv (c rv) c + c d+1 = r v c + c d+1 = r. The last equality ollows rom the deinition o r and the act that v = 1. This implies that is a centerpoint o the points Π (P ) because every hyperplane passing through separates the same set o points as the corresponding hy- perplane through c, which is a centerpoint by deinition. Figure : The stereographic projection o the intersection o the ellipse and a plane through the ocal point is the same as the second ocal point is moved up to ininity. Figure 1: Three examples o E R are illustrated or R = 1, R =, and R = rom let to right. In each case, the stereographic map o a single point is illustrated. For the last case, the pole is at ininity, so the result is a vertical projection. 4 Equivalence o Stereographic Maps Let = [ 1. Let R be a real number and consider the ellipsoid E R R d+1 deined as the points p = [ p p d+1 such that p R 1 + (p d+1 R) R = 1. Theorem 3 Let α and β be real numbers in [1, and let R d+1 be any point. I H is a hyperplane containing, then (Π α ) 1 (Π α (Rd ) H) = (Π β ) 1 (Π β (Rd ) H). Proo. The sphere S α = (Π α ) 1 (Π α (Rd ) H) can also be written as S α = {p R d v (Π α (p) ) = }, where v is the normal vector o H. By translating the points in R d and scaling the space uniormly, we may assume that = [ 1. Let R be any real number in the range [1, and let q R d be any point. We will show that Π R (q) lies on a line through that does not

5 CCCG 13, Waterloo, Ontario, August 8 1, 13 depend on R. Thus, v (Π α (q) ) = i and only i v (Π β (q) ) = and thereore, the spheres Sα and S β must be equal or all values o α and β. Let p = [ p p d+1 be the projection Π R(q). All o the relevant points,, p, and q are contained in a plane, so we proceed in two dimensions, letting x = p, y = p d+1, and a = q. x Since p lies on E R, we have R 1 + (y R) R = 1, or equivalently, x R + y(y R)(R 1) =. (1) Since p is also on the line rom [ R, the north pole o E R, to [ q, its planar coordinates x and y satisy the equation y = R a x + R. It ollows that R = ay a(y 1) + x, R 1 =, (a x) a x and y R = xy a x Plugging these values into (1) gives the ollowing. x (ay) y( xy)(a(y 1) + x) + 4(a x) (a x) =. Multiplying by 4(a x) xy and collecting terms yields the line (a 4)x 4a(y 1) =. We observe that this is the equation o a line through = (, 1) as desired. 5 Concluding Remarks The main story o this paper is not entirely new to computational geometry. When the parabolic liting map was irst introduced in the problem o computing Voronoi diagrams by Edelsbrunner and Seidel [4, they replaced previous methods based on the stereographic map. Today, the parabolic liting is the preerred method or reducing Delaunay triangulations to convex hulls in one higher dimension. Throughout, we have worked directly with the Euclidean coordinates. This gave the beneit o making the computations more immediately clear, but came at the cost o considering several special cases at ininity. An alternative approach would be to exploit the power o projective geometry, which has been shown to be the natural language or stereographic projections. Even the parabola is just an ellipse in the projective plane. For more on projective geometry, I highly recommend the book by Richter-Gebert [17. I would like to thank Marc Glisse or helpul conversations and advice on high school algebra. Reerences [1 D. K. Blandord, G. E. Blelloch, D. E. Cardoze, and C. Kadow. Compact representations o simplicial meshes in two and three dimensions. Int. J. Comput. Geometry Appl., 15(1):3 4, 5. [ T. M. Chan. Polynomial-time approximation schemes or packing and piercing at objects. J. Algorithms, 46(): , 3. [3 K. Clarkson, D. Eppstein, G. Miller, C. Sturtivant, and S.-H. Teng. Approximating center points with iterated Radon points. International Journal o Computational Geometry and Applications, 6(3): , Sep invited submission. [4 H. Edelsbrunner and R. Seidel. Voronoi diagrams and arrangements. Discrete & Computational Geometry, 1(1):5 44, [5 D. Eppstein, G. L. Miller, and S.-H. Teng. A deterministic linear time algorithm or geometric separators and its applications. Fundamenta Inormatica, (4):39 39, [6 S. Har-Peled. A simple proo o the existence o a planar separator, July [7 R. J. Lipton, D. J. Rose, and R. E. Tarjan. Generalized nested dissection. SIAM Journal on Numerical Analysis, 16(): , [8 R. J. Lipton and R. E. Tarjan. A separator theorem or planar graphs. SIAM J. Appl. Math, 36: , [9 G. L. Miller and D. R. Sheehy. Approximate centerpoints with proos. Computational Geometry: Theory and Applications, 43(8): , 1. [1 G. L. Miller, S.-H. Teng, W. Thurston, and S. A. Vavasis. Automatic mesh partitioning. In A. George, J. Gilbert, and J. Liu, editors, Graphs Theory and Sparse Matrix Computation, The IMA Volumes in Mathematics and its Application, pages Springer-Verlag, Vol 56. [11 G. L. Miller, S.-H. Teng, W. Thurston, and S. A. Vavasis. Separators or sphere-packings and nearest neighborhood graphs. Journal o the ACM, 44(1):1 9, [1 G. L. Miller, S.-H. Teng, W. Thurston, and S. A. Vavasis. Geometric separators or inite-element meshes. SIAM J. Comput., 19(): , [13 G. L. Miller, S.-H. Teng, and S. A. Vavasis. A uniied geometric approach to graph separators. In Proceedings o the 3nd Annual IEEE Symposium on Foundations o Computer Science, pages , [14 G. L. Miller and W. Thurston. Separators in two and three dimensions. In Proceedings o the th Annual ACM Symposium on Theory o Computing, pages ACM, 199. [15 G. L. Miller and S. A. Vavasis. Density graphs and separators. In Proceedings o the Second Annual ACM- SIAM Symposium on Discrete Algorithms, pages ACM-SIAM, 1991.

6 5 th Canadian Conerence on Computational Geometry, 13 [16 W. Mulzer and D. Werner. Approximating Tverberg points in linear time or any ixed dimension. In Proceedings o the 8th ACM Symposium on Computational Geometry, pages 33 31, 1. [17 J. Richter-Gebert. Perspectives on Projective Geometry. Springer, 11. [18 W. D. Smith and N. C. Wormald. Geometric separator theorems and applications. In Proceedings o the 39th Annual IEEE Symposium on Foundations o Computer Science, pages 3 43, 1998.