S. Smorodinsky, J.S.B Mitchell, and M. Sharir. Sharp bounds on geometric permutations of pairwise disjoint balls in R . Discrete Comput. Geom., 23(2):247-259, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....size n in R d . Then, it is known that g 2 (n) 2n 2 [2] and that g d (n) is in n d 1 ) 4] and O(n 2d 2 ) 7] Thus, for dimensions d 3, there is a substantial gap between the known upper and lower bounds on geometric permutations. This gap motivated Smorodinsky, Mitchell, and Sharir [5] to consider a natural special case of convex bodies: balls. They proved that the maximum number of geometric permutations achieved by n pairwise disjoint balls in R d is O(n d 1 ) Moreover, this bound is tight as they were able to show a matching lower bound construction. An interesting ....

....S are congruent (say, unit radius balls) It was conjectured that for unit radius balls, the number of geometric permutations drops precipitously to O(1) with the constant depending on d. The planar case of this conjecture was proved by Katchalski and Asinowski [1] and also by Smorodinsky et al. [5]. They showed that when jSj is suciently large, the maximum number of geometric permutations admitted by n unit disks is two. Recently, we proved the unit radius ball conjecture for arbitrary dimensions, and showed that for n congruent balls in R d , the maximum number of geometric permutations ....

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Shakhar Smorodinsky, Joseph S.B. Mitchell, and Micha Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in R d . Discrete and Computational Geometry, 23:247-259, 2000.

....g d (n) have been known for some time: 1. g 2 (n) 2n 2 [2] 2. g d (n) n d 1 ) 4] 3. g d (n) O(n 2d 2 ) 7] Thus, for dimensions d 3, there is a substantial gap between the known upper and lower bounds on geometric permutations. This gap motivated Smorodinsky, Mitchell, and Sharir [5] to consider a natural special case of convex bodies: balls. They proved that the maximum number of geometric permutations achieved by n pairwise disjoint balls in R d is O(n d 1 ) Moreover, this bound is tight as they were able to show a matching lower bound construction. An interesting ....

....S are congruent (say, unit radius balls) It is conjectured that for unit radius balls, the number of geometric permutations drops precipitously to O(1) with the constant depending on d. The planar case of this conjecture was proved by Katchalski and Asinowski [1] and also by Smorodinsky et al. [5]. They showed that 1 when jSj is suciently large, the maximum number of geometric permutations admitted by n unit disks is two. Recently, we proved the unit radius ball conjecture for arbitrary dimensions, and showed that for n congruent balls in R d , the maximum number of geometric ....

[Article contains additional citation context not shown here]

Shakhar Smorodinsky, Joseph S.B. Mitchell, and Micha Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in R d . Discrete and Computational Geometry, 23:247{ 259, 2000.

.... established about a decade ago: the number is in Omega Gamma n d Gamma1 ) KLL92] and O(n 2d Gamma2 ) Wen90] with a tight bound of 2n Gamma 2 known for d = 2 [ES90] The study of geometric permutations was revitalized by a focus specifically on balls by Smorodinsky, Mitchell, and Sharir [SMS00]. They closed the gap in this special case to Theta(n d Gamma1 ) a result which was quickly followed by an extension to the same bound for fat convex objects [KV99] a) b) 1 2 3 4 5 Figure 1: a) Two line traversals with permutations (1; 2; 3; 4; 5) and (1; 2; 4; 3; 5) b) Three ....

....result which was quickly followed by an extension to the same bound for fat convex objects [KV99] a) b) 1 2 3 4 5 Figure 1: a) Two line traversals with permutations (1; 2; 3; 4; 5) and (1; 2; 4; 3; 5) b) Three permutations achievable. One fascinating line of investigation opened in [SMS00], and independently by Asinowski and Katchalski [Asi98] is the even more special case of congruent balls. They proved that sufficiently many congruent disks in R 2 admit only a constant number of geometric permutations in fact, just 2. A little experimentation (Fig. 1a) quickly reveals the ....

S. Smorodinsky, J.S.B. Mitchell, and M. Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in R d . Discrete Comput. Geom., 23:247--259, 2000.

....R d can realize n d 1 ) geometric permutations. The best upper bound known for general convex bodies is O(n 2d 2 ) by Wenger [11] Thus, there remains a substantial gap between the known upper and lower bounds on geometric permutations. This gap motivated Smorodinsky, Mitchell, and Sharir [9] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. 9] were able to prove a matching asymptotic upper bound, thus showing that 1 2 the maximum number of ....

....upper and lower bounds on geometric permutations. This gap motivated Smorodinsky, Mitchell, and Sharir [9] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. [9] were able to prove a matching asymptotic upper bound, thus showing that 1 2 the maximum number of geometric permutations for collections of n disjoint balls in R d is (n d 1 ) An interesting special case of the problem is when all balls in S are congruent (say, unit balls) Smorodinsky ....

[Article contains additional citation context not shown here]

S. Smorodinsky, J. S.B. Mitchell, and M. Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in R d . Discrete and Computational Geometry, 23:247-259, 2000.

....bounds on geometric permutations. 1 (n 1, 1, 2, n 2, n 1) n n 1 n 2 n 2 segments (1, 2, n 2, n 1, n) 1, 2, n 2, n, n 1) 1 Figure 1: An example of n convex objects admitting 2n 2 geometric permutations. borrowed from [8] This gap motivated Smorodinsky, Mitchell, and Sharir [9] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. 9] were able to prove a matching asymptotic upper bound, thus showing that the maximum number of geometric ....

....2n 2 geometric permutations. borrowed from [8] This gap motivated Smorodinsky, Mitchell, and Sharir [9] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. [9] were able to prove a matching asymptotic upper bound, thus showing that the maximum number of geometric permutations for collections of n disjoint balls in R d is (n d 1 ) An interesting special case of the problem is when all balls in S are congruent (say, unit balls) Smorodinsky et al. ....

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S. Smorodinsky, J. S.B. Mitchell, and M. Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in r d . Discrete and Computational Geometry, 23:247-259, 2000.

....3, there is a large gap between the known upper and lower bounds. It has been conjectured that g d (n) O(n d Gamma1 ) For pairwise disjoint spheres in R d , better results have been given. For a set A of n congruent spheres (i.e. all radii are the same) Smorodinsky, Mitchell, and Sharir [7] proved that A in R 2 admits only two different geometric permutations if n is sufficiently large (this fact was also independently discovered by Katchalski and Asinowski [2] they conjectured that the maximum number of geometric permutations for A in R d is O(1) for d 3. Recently, Zhou and ....

....they conjectured that the maximum number of geometric permutations for A in R d is O(1) for d 3. Recently, Zhou and Suri [8] proposed a proof for this conjecture, showing that the maximum number of such geometric permutations is 16. For n noncongruent spheres in R d , Smorodinsky et al. [7] Dept. of Comp. Sci. Eng. Univ. of Notre Dame, Notre Dame, IN 46556, USA. fyhuang3, cheng cse.nd.edu. This research was supported in part by the National Science Foundation under Grants CCR 9623585 and CCR 9988468. y Dept. of Comp. Sci. Eng. State Univ. of New York at Buffalo, Bell Hall ....

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S. Smorodinsky, J.S.B. Mitchell, and M. Sharir, "Sharp bounds on geometric permutations of pairwise disjoint balls in R d ," Discrete Comput. Geom., 23:247-259, 2000.

....(1, 2, n 2, n 1, n) 1, 2, n 2, n, n 1) 1 Figure 7.1: An example of n convex objects admitting 2n 2 geometric permutations. Taken from [30] It consists of n 2 segments labeled 1; 2; n 2 and two close disks labeled n 1; n. This gap motivated Smorodinsky, Mitchell, and Sharir [42] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. 42] were able to prove a matching asymptotic upper bound, thus showing that the maximum number of geometric ....

....1; 2; n 2 and two close disks labeled n 1; n. This gap motivated Smorodinsky, Mitchell, and Sharir [42] to consider a natural, but specialized, family of convex bodies: balls. It turns out that even for families of balls n d 1 ) geometric permutations are realizable. Smorodinsky et al. [42] were able to prove a matching asymptotic upper bound, thus showing that the maximum number of geometric permutations for collections of n disjoint balls in R d is (n d 1 ) An interesting special case of the problem is when all balls in S are congruent (say, unit balls) Smorodinsky et al. ....

[Article contains additional citation context not shown here]

Shakhar Smorodinsky, Joseph S.B. Mitchell, and Micha Sharir. Sharp bounds on geometric permutations for pairwise disjoint balls in R d . Discrete and Computational Geometry, 23:247-259, 2000.

....d Matthew J. Katz Kasturi R. Varadarajan y November 23, 1999 Abstract We show that the maximum number of geometric permutations of a set of n fat pairwise disjoint convex objects in R d is O(n d Gamma1 ) This generalizes the bound of Theta(n d Gamma1 ) obtained by Smorodinsky et al. [4] on the number of geometric permutations of n pairwise disjoint balls. Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva 84105, Israel. E mail: matya cs.bgu.ac.il. Work by this author has been supported by the Israel Science Foundation founded by the ....

....of A, and g d (n) the maximum of g d (A) over all families A of pairwise disjoint convex sets. The study of geometric permutations was initiated by Katchalski et al. 2] The following bounds are known for g d (n) For a more detailed summary of the literature, the reader is referred to [4]. 1. g 2 (n) 2n Gamma 2 (Edelsbrunner and Sharir [1] 2. g d (n) Omega Gamma n d Gamma1 ) Katchalski et al. 3] 3. g d (n) O(n 2d Gamma2 ) Wenger [5] Thus for d 2, there is a large gap between the upper and lower bounds, and bridging the gap seems challenging. Smorodinsky et al. ....

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S. Smorodinsky, J.S.B. Mitchell and M. Sharir. Sharp bounds on geometric permutations of pairwise disjoint balls in R d . Proc. 15th ACM Symp. on Computational Geometry, 1999.

....[4] have shown that g 2 (n) 2n 2. Katchalski et al. 6] showed that g d (n) n ) The only known general upper bound on g d (n) is O(n ) and is due to Wenger [5] Hence, for d 3 there still exists a wide gap between the known upper and lower bounds. Recently, Smorodinsky et al. [8, 9], obtained the sharp upper and lower bounds of (n ) on the number of geometric permutations, in the special case where A consists of pairwise disjoint balls in R . For the case of congruent balls, or nearly congruent the number of geometric permutations is only O(1) 11] This result was ....

....was a Ph.D. student under the supervision of Micha Sharir. 1 to the case of fat convex bodies, by Katz and Varadarajan [10] It has been conjectured that g d (n) is O(n ) 1. 1 Separation Sets and Neighbors Wenger [5] introduced the notion of separation set which was later generalized in [9]: De nition 1.1 Let S be a family of pairwise disjoint convex sets in R , and let P be a set of hyperplanes in R passing through the origin. We say that P is a separation set for S if for each pair s i ; s j 2 S there exists a hyperplane h, parallel to a hyperplane in P , such that s i and ....

[Article contains additional citation context not shown here]

S. Smorodinsky, J.S.B Mitchell, and M. Sharir. Sharp bounds on geometric permutations of pairwise disjoint balls in R . Discrete Comput. Geom., 23(2):247-259, 2000.

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