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On Neighbors in Geometric Permutations

On Neighbors in Geometric Permutations. Shakhar Smorodinsky Tel-Aviv University Joint work with Micha Sharir. Geometric Permutations. A - a set of disjoint convex bodies in R d A line transversal l of A induces a geometric permutation of A. l 2 : <2,3,1>. l 1 : <1,2,3>. l 2.

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On Neighbors in Geometric Permutations

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  1. On Neighbors in Geometric Permutations Shakhar Smorodinsky Tel-Aviv University Joint work with Micha Sharir

  2. Geometric Permutations • A - a set of disjoint convex bodies in Rd • A line transversal l of A induces a geometric permutation of A l2: <2,3,1> l1: <1,2,3> l2 2 3 1 l1

  3. 1 2 3 Example <2,1,3> <2,3,1> <1,2,3>

  4. 3 2 n-2 1 An example of S with2n-2 geometric permutations(Katchalski, Lewis, Zaks - 1985) <2,3,…,n-2,1> <3,..2,…,n-2,1>

  5. Motivation? Trust me ……………. There is some!

  6. Problem Statement gd(A) = the number of geometric permutations of A gd(n) = max|A|=n{gd(A)} ? < gd(n) < ?

  7. Known Facts • g2(n) = 2n-2 (Edelsbrunner, Sharir 1990) • gd(n) = (nd-1) (Katchalski, Lewis, Liu 1992) • gd(n) = O(n2d-2) (Wenger 1990) • Special cases: • narbitraryballs in Rd have at most (nd-1)GP’s (Smorodinsky, Mitchell, Sharir 1999) • The (nd-1) boundextended to fat objects (Katz, Varadarajan 2001) • nunit balls in Rd have at mostO(1) GP’s (Zhou, Suri 2001) 4

  8. Overview of our result • Define notion of “Neighbors” • Neighbors Lemma: few neighbors => few permutations • In the plane (d = 2) few neighbors • Conjecture: few neighbors in higher dimensions (d > 2)

  9. bi bj Definition A- a set of convex bodies in Rd A pair (bi, bj) in A are called neighbors If  geometric permutation for which bi, bj appear consecutive:

  10. 3 2 n-2 1 Neighbors Example (2,3) (2,4) ….. (2,n-2)

  11. Neighbors No neighbors !!!

  12. h b1 b2 Neighbors Lemma In Rd, if Nis the set of neighbor pairs of A, then gd(A)=O(|N|d-1). Proof: Fix a neighbor pair (b1,b2) in N. Unit Sphere Sd-1 h b1 . b2 b1 is crossed before b2

  13. Neighbors Lemma (cont) In Rd, if Nis the set of neighbor pairs of A, then gd(A)=O(|N|d-1). Let P be a set of |N| separating hyperplanes (A hyperplane for each one of the neighbor pairs)

  14. Consider the arrangement of great circles that correspond to hyperplanes in P. connected component C => Unique GP A fixed permutation in C Unit Sphere Sd-1 C # connected component <O(|N|d-1)

  15. The # neighbors of n convex bodies in Rd is O(n) Conjecture: If true, implies a (nd-1)upper bound on gd(n)! Else, Return (to SWAT 2004);

  16. 1 5 4 2 3 O(n) upper bound on the # neighbors inthe plane Upper Bound: #neighbors in the plane Construct a “neighbors graph”

  17. 1 5 4 2 3 Connect neighbors as follows: Neighbors Graph (cont) Inthis drawing rule there may existcrossings

  18. Neighbors Graph (cont) : • However: there are no three pairwise crossing edges (technical proof) A graph that can be drawn in the plane with no three pairwise crossing edges is called a quasi-planar graph. Theorem: [Agarwal et al. 1997] A quasi-planar graph with nvertices has O(n) edges.

  19. Further research • Prove: O(n) neighbors in any fixed dimension. • In the plane: Is the neighbors graphplanar?

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