C. Rogers, Covering a sphere with spheres, Mathematika 10 (1963) 157--164. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R such that U = fw=kwk : w 2 Wg is an net in S . It is clear from the de nition of W that for each w 2 W we have hwi = O(h i) The order of in the previous lemma is optimal, but the proof does not provide the bestknown constant, which apparently can be deduced from a result of Rogers [24]. We can now state the main result. In an oracle polynomial time algorithm (see [15, p. 27] each call of the oracle counts one step. Of course, since the output of the oracle is used in the algorithm, its time depends also on the size of the unknown polytope P . Theorem 7.2. For each xed n 2 N ....

C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963), 157-164.

....is a set of O(# ) rational vectors in R . It is clear from the definition of W that for each w W we have #w#=O(###) The order of # in the previous lemma is optimal, but the proof does not provide the best known constant, which apparently can be deduced from a result of Rogers [24]. We can now state the main result. In an oracle polynomial time algorithm (see p. 27 of [15] each call of the oracle counts one step. Of course, since the output of the oracle is used in the algorithm, its time depends also on the size of the unknown polytope P . Theorem 7.2. For each fixed ....

C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963), 157--164.

.... is a set of O(z ) rational vectors in R such that v = w llwll:w is net in It is clear from the definition of W that for each w w (w) The order of z in the previous lemma is optimal, but the proof does not provide the best known constant, which apparently can be deduced from a result of Rogers [24]. We can now state the main result. In an oracle polynomial time algorithm (see [15, p. 27] each call of the oracle counts one step. Of course, since the output of the oracle is used in the algorithm, its time depends also on the size of the unknown polytope P. Theorem 7.2. For each fixed n N, ....

C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963), 157-164.

....distances between points. The image U(n; of Y under this map is therefore an net in S n Gamma1 containing O( 1 Gamman ) points. The order of in the previous lemma is optimal, but the proof does not provide the bestknown constant, which apparently can be deduced from a result of Rogers [23]. We thank Helmut Groemer for providing the proof, much shorter than our original one, to the following lemma. Lemma 7.2. Let 0 r R, let 0 r= 3R) and let U be an net in S n Gamma1 . If K is a convex body in S n Gamma1 such that r hK (u) R for each u 2 U , then r 2 B ae K ae ....

C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963), 157--164.

...., we conclude that E X ThetaX [ae(x; y) is at most B p (log n) n Delta Prob h ae(x; y) B p (log n) n i = B p (log n) n O(n Gamma2 ) O( p (log n) n ) This concludes the proof of the theorem. 2 4 Proof of Lemma 3.2 Proof. The proof is essentially the same as in Rogers [18] and Erdos and Rogers [8] Since we have no explicit reference for the exact result we need here, we include a proof. The calculations are somewhat simpler than those in [18, 8] since a less precise bound is sufficient for our application. In the sequel, we may suppose that n is sufficiently ....

....) This concludes the proof of the theorem. 2 4 Proof of Lemma 3.2 Proof. The proof is essentially the same as in Rogers [18] and Erdos and Rogers [8] Since we have no explicit reference for the exact result we need here, we include a proof. The calculations are somewhat simpler than those in [18, 8], since a less precise bound is sufficient for our application. In the sequel, we may suppose that n is sufficiently large. Let r be given, and set j = r=n 2 . Let v x denote, as before, the n measure of a spherical cap of geodesic radius x. Define N as the smallest integer such that N Delta ....

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C. A. Rogers. Covering a sphere with spheres. Mathematika, 10:157--164, 1963.

....sphere, short the unit (d Gamma 1) sphere. Let 0 be an angle and a be a point of S d Gamma1 . Then the set fx 2 S d Gamma1 : OE(a; x) g is called a spherical cap with center a and angular radius . Remember, OE(a; x) denotes the angle between the halflines fa : 0g and fx : 0g. In [70] Rogers proved the following. Lemma 2.16 [70] Let 0 2 . Then S d Gamma1 can be covered by O(d 3=2 log( d sin ) sin Gamma(d Gamma1) spherical caps with angular radius . Now we define the Delaunay triangulation and the Voronoi diagram for a set P of m points in IR j . ....

....0 be an angle and a be a point of S d Gamma1 . Then the set fx 2 S d Gamma1 : OE(a; x) g is called a spherical cap with center a and angular radius . Remember, OE(a; x) denotes the angle between the halflines fa : 0g and fx : 0g. In [70] Rogers proved the following. Lemma 2. 16 [70] Let 0 2 . Then S d Gamma1 can be covered by O(d 3=2 log( d sin ) sin Gamma(d Gamma1) spherical caps with angular radius . Now we define the Delaunay triangulation and the Voronoi diagram for a set P of m points in IR j . Then we transfer the definition to a set of ....

C. A. Rogers. Covering a sphere with spheres. Mathematika, 10:157--164, 1963.

....insertions or deletions in the hash table, which needs constant time, w.h.p. 5 Extensions to higher dimensional spaces fl angle graphs can be generalized to scenes in IR d . In this case, we have to partition the space around a position x into sectors that are cones with maximum angle fl. In [24] Rogers has shown that k(fl) O(d 3=2 log d sin(fl=2) sin Gammad fl 2 ) many such cones suffice. These cones cover IR d , but overlap. It can be show that the fact that they overlap does not cause a problem for our data structure. We can get similar results to what we have for 2D, the ....

C.A. Rogers. Covering a Sphere with Spheres. Mathematika, 10:157 -- 164, 1963.

....such that two distinct points different from P in the same cone form, at P , an angle at most ff. We use in Theorem 4.2 the well known fact that B(d; ff) is finite for every ff 0 and every d. This covering problem has been extensively studied. We mention the following upper bound due to Rogers [19]: B(m; ff) is O m 3=2 log m sin(ff=2) 1 sin(ff=2) m : Theorem 4.2. Fix m and a norm N on IR m . Let G = V; E) be an n vertex geometric graph in IR m under norm N . Let OPT be the weight of the optimal tour on G. Let T 0 be any 2 optimal tour of G. Then the weight of ....

C. A. Rogers, "Covering a Sphere With Spheres," Mathematika 10 (1963), 157-164.

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C. Rogers, Covering a sphere with spheres, Mathematika 10 (1963) 157--164. 10

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Rogers, C.A. (1963) Covering a sphere with spheres. Mathematika, 10, 157-164.

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C.A. Rogers: "Covering a Sphere with Spheres", Mathematika 10, 1963, 157164.

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C.A. Rogers: "Covering a Sphere with Spheres", Mathematika 10, 1963, 157-164.

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C.A. Rogers: "Covering a Sphere with Spheres", Mathematika 10 1963, 157-164

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C.A. Rogers. Covering a sphere with spheres. Mathematika, 10:157--164, 1963.

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Rogers, C.A., Covering a sphere with spheres, Mathematika 10 (1963), 157-164.

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