Almost Tight Bound for a Single Cell in an Arrangement of Convex Polyhedra in
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Almost Tight Bound for a Single Cell in an Arrangement of Convex Polyhedra in R 3 Esther Ezra Tel-Aviv University. A single cell of an arrangement of convex polyhedra. Input:  = {P 1 , …, P k } a collection of k convex polyhedra in 3-space with n facets in total.

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Almost Tight Bound for a Single Cell in an Arrangement of Convex Polyhedra in R3EstherEzra Tel-Aviv University


A single cell of an arrangement of convex polyhedra
A single cell of an arrangement of convex polyhedra Convex Polyhedra in

Input:

 = {P1, …, Pk} a collection of k convex polyhedra in 3-space with n facets in total.

A( ) : The arrangement induced by .

The problem

What is the maximal number of vertices/edges/faces that form the boundary of a single cell of A( ) ?

Combinatorial complexity


Motivation translational motion planning
Motivation: Convex Polyhedra in Translational motion planning

Input

Robot R , a set A = {A1, …, Ak} of k disjoint obstacles.

The free space

The set of all legal placements of R.

The workspace

collision

R does not intersect any of the obstacles in A


The configuration space
The configuration space Convex Polyhedra in

The robot R is mapped to a point.

Each obstacle Ai is mapped to the set:

Pi = { (x,y,z) : R(x,y,z)  Ai  } = Ai(-R(0,0,0))

A point p in Pi corresponds to an illegal placement of Rand vice versa.

The forbidden placements of R

The Minkowski sum

The expanded obstacle


The free space
The free space Convex Polyhedra in

The free space is

An algorithm that constructs the union?

Not efficient when the complexity of the whole union is high (cubic).


Restriction a single component of the free space
Restriction: Convex Polyhedra in A single component of the free space

A single component of

The subset of all placements reachable from a given initial free placement of R via a collision-free motion.


Restatement a single component in the complement of the union
Restatement: Convex Polyhedra in A single component in the complement of the union

Input

 = {P1, …, Pk} a collection of k convex polyhedra in 3-space with n facets in total.

The problem

What is the maximal number of vertices/edges/faces that formthe boundary of a single component of ?

Minkowski sum of a convex obstacle with a convex part of -R

It is sufficient to bound the number of intersection vertices

A single cell of A( )


Single (bounded) cell in 2D Convex Polyhedra in


The unbounded cell in 3D Convex Polyhedra in

Ω(nk) vertices

Ω(k2) vertices

Can be modified to Ω(nk(k)) vertices


Previous results
Previous results Convex Polyhedra in

  • R2: Aronov & Sharir 1997. Θ(n(k)) .

  • R3:Aronov & Sharir 1990. O(n7/3 log n) .

  • Rd :Aronov & Sharir 1994. O(nd-1 log n) .

  • R3:Halperin & Sharir 1995. O(n2+) ,   > 0 .

  • Rd :Basu 2003. O(nd-1+),   > 0 .

    1-4: Comparable algorithmic bounds.

    The case of convex polyhedra in R3:

    Use [Aronov & Sharir 1994] O(n2 log n) .

    This bound does not depend on k.

Many components

Curved simply-shaped regions

Simply-shaped regions


Our result
Our result Convex Polyhedra in

The combinatorial complexity of a single cell of A( ) is O(nk1+) ,  > 0 .

We use a variant of the technique of [Halperin & Sharir 1995] .

We present a deterministic algorithm that constructs a single cell in O(nk1+ log2 n) time,   > 0 .

The bound depends on the number k of polyhedra

Crucial: The input regions are of constant description complexity


Classification of the intersection vertices
Classification of the intersection vertices Convex Polyhedra in

Outer vertex:The intersection of an edge of a polyhedron with a facet of another polyhedron.Overall number: O(nk) .

Inner vertex:The intersection of three facets of three distinct polyhedra.Overall number: O(nk2) .

u


The combinatorial complexity of the unbounded cell
The combinatorial complexity of the unbounded cell Convex Polyhedra in

How many inner vertices are on the unbounded cell of A( ) ?


Analysis exposed convex chains
Analysis: Exposed convex chains Convex Polyhedra in

Classify each vertex v by: How long can we freely go from v when alternating out-of/into the unbounded cell.

Not meeting any polyhedra

1 step

After the removal of P’: 4 steps


Analysis continue
Analysis: Continue Convex Polyhedra in

We trace this way Exposed convex chains.

Number of steps = length of the chain

V(j)( ) –the number of inner vertices of the unbounded cell of A( ) with j steps.

5 steps

V(0)( ) bounds the overall number of inner vertices of the unbounded cell.


The overall complexity of exposed chains
The overall complexity of exposed chains Convex Polyhedra in

Exposed chains of length 4

Use recurrence: V(j)( )  V(j+1)( )

Exposed chains of length 4 or 5

Lemma:

  • The number of vertices on exposed chains of length  5 isO(nk).

  • The number of vertices on exposed closed chains (of length 4) isO(nk).

Multiply by O(k).

This is the only interesting case.


Solving the recurrence
Solving the recurrence Convex Polyhedra in

V(j)() = O(nk1+) ,  > 0, 0  j  4

The combinatorial complexity of a single cell of A( ) is O(nk1+) ,  > 0 .


Thank you Convex Polyhedra in


The charging scheme case 2
The charging scheme: Case (2) Convex Polyhedra in


Exposed chains of length 5
Exposed chains of length Convex Polyhedra in  5

’

=M  P_3

’=M  P’

’

M=F_1  P_2


Special quadrilateral
Special quadrilateral Convex Polyhedra in


Special vertex
Special vertex Convex Polyhedra in


Union of polyhedra in r 3
Union of polyhedra in Convex Polyhedra in R3

Input:

 = {P1, …, Pk} a collection of k polyhedra in 3-space with n facets in total.

The problem

What is the maximal number of vertices/edges/faces that form the boundary of the (complement of) the union?Trivial upper bound:O(n3) .Lower bound:Ω(n3),for non-convex polyhedra.

Combinatorial complexity.

An algorithm that constructs the union: Not efficient.


The combinatorial problem convex polyhedra
The combinatorial problem: Convex Polyhedra in Convex polyhedra

Motion planning

[Aronov, Sharir 1997]

is a set of convex polyhedra that arise in the case of convex translating robot R Minkowski sums of (-R) and the obstacles:O(nk log k)

Lower bound: (nk(k))

Construction time: O(nk log k log n)

The general problem

[Aronov, Sharir, Tagansky 1997]

is a set of convex polyhedra :O(k3 + nk log k)

Lower bound: (k3 + nk(k))

Construction time: O(k3 + nk log k log n)

Cannot be applied when R is non-convex.


The combinatorial problem non convex polyhedra
The combinatorial problem: Convex Polyhedra in Non-convex polyhedra

is a set of general polyhedra : Θ(n3) .

Also holds in translational motion planning problem.

Not necessarily convex.

k = O(1)


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