Almost Tight Bound for a Single Almost Tight Bound for a Single Cell in an Arrangement of Cell in an Arrangement of Convex Polyhedra in Convex Polyhedra in RR33

EstherEsther Ezra Ezra Tel-Aviv UniversityTel-Aviv University

A single cell of an arrangement of A single cell of an arrangement of convex polyhedraconvex polyhedra

Input:

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

A( ) : The arrangement induced by .

The problemWhat is the maximal number of vertices/edges/faces that

form the boundary of a single cell of A( ) ?

Combinatorial complexity

Motivation: Motivation: Translational motion planningTranslational motion planning

Input

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

The free space

The set of all legal placements of R.

R does not intersect any of the obstacles in

The workspace

collision

The configuration spaceThe configuration space

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 R and vice versa.

The forbidden placements of R

The Minkowski sum

The expanded obstacle

The free spaceThe free space

The free space is

An algorithm that constructs the union?

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

k

i iP1

Restriction:Restriction:A single component of the free spaceA 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.

k

i iP1

Restatement:Restatement: A single component in the A single component in the complement of the unioncomplement 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 ?k

i iP1

It is sufficient to bound the number

of intersection vertices

Minkowski sum of a convex obstacle with a

convex part of -R

A single cell of A( )

Single (bounded) cell in 2DSingle (bounded) cell in 2D

The unbounded cell in 3DThe unbounded cell in 3D

Ω(nk) vertices

Ω(k2) vertices

Can be modified to Ω(nk(k))

vertices

Previous resultsPrevious results

1. R2 : Aronov & Sharir 1997. Θ(n(k)) .2. R3 : Aronov & Sharir 1990. O(n7/3 log n) .3. Rd : Aronov & Sharir 1994. O(nd-1 log n) .4. R3 : Halperin & Sharir 1995. O(n2+) , > 0 .5. 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

Simply-shaped regionsCurved

simply-shaped regions

Our resultOur result

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 Classification of the intersection verticesvertices

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 The combinatorial complexity of the unbounded cellunbounded cell

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

Analysis: Exposed convex chainsAnalysis: Exposed convex chains

Not meeting any polyhedra

After the removal of P’: 4 steps

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

the unbounded cell.

1 step

Analysis: ContinueAnalysis: Continue

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.

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

5 steps

The overall complexity of exposed The overall complexity of exposed chainschains

Exposed chains of length Exposed chains of length 4 4Use recurrence: V(j)() V(j+1)()

Exposed chains of length 4 or 5Exposed chains of length 4 or 5Lemma:• The number of vertices on exposed chains of length 5

is O(nk) .• The number of vertices on exposed

closed chains (of length 4) is O(nk) .

Multiply by O(k).

This is the only interesting case.

Solving the recurrenceSolving the recurrence

V(j)() = O(nk1+) , > 0, 0 j 4

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

Thank youThank you

The charging scheme: Case (2)The charging scheme: Case (2)

Exposed chains of length Exposed chains of length 5 5

M=F_1 P_2

’=M P’=M P_3

’

’

Special quadrilateralSpecial quadrilateral

Special vertexSpecial vertex

Combinatorial complexity.

Union of polyhedra in Union of polyhedra in RR33

Input: = {P1, …, Pk} a collection of k polyhedra in 3-space

with n facets in total.

The problemWhat 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.

An algorithm that constructs the union: Not efficient.

The combinatorial problem: The combinatorial problem: Convex polyhedraConvex 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: The combinatorial problem: Non-convex polyhedraNon-convex polyhedra

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

k = O(1)

Also holds in translational motion planning problem.

Not necessarily convex.