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 Rd

A line transversal l of A induces a geometric permutation of A

l2

l11

2 3

l1: <1,2,3> l2: <2,3,1>

3

Example

3

1

2

<1,2,3>

<2,3,1>

<2,1,3>

An example of S with 2n-2 geometric permutations(Katchalski, Lewis, Zaks - 1985)

1

<2,3,…,n-2,1><3,..2,…,n-2,1>

2

3

n-2

Motivation?

Trust me …………….

There is some!

Problem Statement

gd(A) = the number of geometric permutations of A

gd(n) = max|A|=n{gd(A)}

? < gd(n) < ?

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: n arbitrary balls in Rd have at most (nd-1) GP’s

(Smorodinsky, Mitchell, Sharir 1999) The (nd-1) bound extended to fat objects (Katz, Varadarajan 2001) n unit balls in Rd have at most O(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)

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:

bi bj

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Neighbors

1 2

3

n-2

Example

(2,3)

(2,4)

…..

(2,n-2)

11

Neighbors

No neighbors !!!

Neighbors Lemma

In Rd, if N is the set of neighbor pairs

of A, then gd(A)=O(|N|d-1).

b1 b2

hb1

b2.

hUnit Sphere

Sd-1

b1 is crossed before b2

Proof: Fix a neighbor pair (b1,b2) in N.

Neighbors Lemma (cont)

In Rd, if N is 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)

Consider the arrangement of great circles

that correspond to hyperplanes in P.

A fixed permutation in C

C

connected component C => Unique GP

Unit Sphere

Sd-1

# connected component < O(|N|d-1)

Conjecture:

The # neighbors of n convex bodies in Rd is O(n)

If true,

implies a (nd-1) upper bound on gd(n)!

Else,

Return (to SWAT 2004);

Upper Bound: #neighbors in the plane

O(n) upper bound on the # neighbors in the plane

Construct a “neighbors graph”

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1

4

2

5

Neighbors Graph (cont)

3

1

4

2

5

Connect neighbors as follows:

In this drawing rule there may exist crossings

Neighbors Graph (cont) : However: there are no three pairwise crossing

edges (technical proof)

Theorem: [Agarwal et al. 1997]

A quasi-planar graph with n vertices has O(n) edges.

A graph that can be drawn in the plane with no three pairwise crossing edges is called

a quasi-planar graph.

Further research Prove: O(n) neighbors in any

fixed dimension. In the plane:

Is the neighbors graph planar?