Voronoi Diagrams, Delaunay Triangulations and Polytopes

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Voronoi Diagrams, Delaunay Triangulations and Polytopes

Jean-Daniel Boissonnat

MPRI, Lecture 2

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Voronoi diagrams in nature

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The solar system (Descartes)

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Growth of merystem

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Euclidean Voronoi diagrams

Voronoi cell V(pi) ={x:kx−pik ≤ kx−pjk, ∀j}

Voronoi diagram(P) ={collection of all cellsV(pi),p_i ∈P}

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Voronoi diagrams and polytopes

Polytope

The intersection of a finite collection of half-spaces : V =T

i∈I h⁺_i

Each Voronoi cell is a polytope

The Voronoi diagram has the structure of acellcomplex

The Voronoi diagram ofPis the projection of a polytope ofR^d+1

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Voronoi diagrams and polytopes

Polytope

i∈I h⁺_i

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Voronoi diagrams and polytopes

Polytope

i∈I h⁺_i

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Voronoi diagrams and polytopes

Polytope

i∈I h⁺_i

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Voronoi diagrams and polyhedra

Vor(p1, . . . ,pn)is theminimization diagram of thenfunctionsδi(x) = (x−pi)²

arg min(δi) =arg max(hi) wherehpi(x) =2pi·x−p²i

The minimization diagram of theδiis also the maximization diagram of theaffine functionshpi(x)

The faces of Vor(P)are the projections of the faces ofV(P) =T

i h⁺_p_i h⁺pi ={x:xd+1>2pi·x−p²i}

p_i

z= (x−p_i)²

Note !

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Voronoi diagrams and polyhedra

p_i

z= (x−p_i)²

Note !

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Voronoi diagrams and polyhedra

Note !

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Voronoi diagrams and polyhedra

i h⁺_p_i

h⁺pi ={x:xd+1>2pi·x−p²i}

Note !

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Voronoi diagrams and polyhedra

i h⁺_p_i

h⁺pi ={x:xd+1>2pi·x−p²i}

Note !

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Voronoi diagrams and polytopes

Lifting map

The faces ofVor(P)are the projection of the faces of the polytope V(P) =T

i h⁺_p_i

whereh_p_i is the hyperplane tangent to paraboloidQat the lifted point(pi,p²_i)

Corollaries

I The size ofVor(P)is the same as the size ofV(P)

I ComputingVor(P)reduces to computingV(P)

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Voronoi diagrams and polytopes

Lifting map

The faces ofVor(P)are the projection of the faces of the polytope V(P) =T

i h⁺_p_i

whereh_p_i is the hyperplane tangent to paraboloidQat the lifted point(pi,p²_i)

Corollaries

I The size ofVor(P)is the same as the size ofV(P)

I ComputingVor(P)reduces to computingV(P)

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Polytopes (convex polyhedra)

Two ways of defining polytopes

Convex hull of a finite set of points : V =conv(P)

Intersection of a finite set of half-spaces: H=∩^h∈Hh⁺_i

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Facial structure of a polytope

Supporting hyperplaneh: H∩ P 6=∅

P on one side ofh Faces:P ∩h,hsupp. hyp.

Dimension of a face: the dim. of its affine hull

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General position

Points in general position

I Pis in general position iff no subset ofk+2points lie in ak-flat

⇒ IfPis in general position, all faces ofconv(P)are simplices Hyperplanes in general position

I His in general position iff the intersection of any subset ofd−k hyperplanes intersect in ak-flat

⇒ anyk-face is the intersection ofd−khyperplanes

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General position

⇒ IfPis in general position, all faces ofconv(P)are simplices

Hyperplanes in general position

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General position

⇒ IfPis in general position, all faces ofconv(P)are simplices

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Duality between points and hyperplanes

hyperplane ofR^d h:xd =a·x⁰−b −→ point h^∗= (a,b)∈R^d

point p= (p⁰,pd)∈R^d −→ hyperplane p^∗⊂R^d

={(a,b)∈R^d:b=p⁰·a−pd}

Duality

I preserves incidences:

p∈h ⇐⇒ pd =a·p⁰−b⇐⇒b=p⁰·a−pd⇐⇒h^∗ ∈p^∗ p∈h⁺ ⇐⇒ p_d >a·p⁰−b⇐⇒b>p⁰·a−p_d⇐⇒h^∗ ∈p^∗+

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Duality between points and hyperplanes

hyperplane ofR^d h:xd =a·x⁰−b −→ point h^∗= (a,b)∈R^d

point p= (p⁰,pd)∈R^d −→ hyperplane p^∗⊂R^d

={(a,b)∈R^d:b=p⁰·a−pd}

Duality

I preserves incidences:

p∈h ⇐⇒ pd =a·p⁰−b⇐⇒b=p⁰·a−pd⇐⇒h^∗ ∈p^∗ p∈h⁺ ⇐⇒ p_d >a·p⁰−b⇐⇒b>p⁰·a−p_d⇐⇒h^∗ ∈p^∗+

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Duality between polytopes

Leth1, . . . ,hn benhyperplanes ofR^dand letH=∩h⁺_i

ss

h₁

h2 *

h3

h^∗₃

h^∗₂ h^∗₁

A vertexsofHis ¯the intersection ofk≥d hyperplanesh1, . . . ,hklying above all the other hyperplanes

=⇒s^∗is a hyperplane that 1. containsh^∗₁, . . . ,h^∗_k

2. supports H^∗= conv⁻(h^∗₁, . . . ,h^∗_k)

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More generally and under the general position assumption, Letf be a(d−k)-face ofH and aff(f) =∩^ki=1h_i

p∈f ⇔ h^∗i ∈p^∗ for i=1, . . . ,k h^∗i ∈p^∗+ for i=k+1, . . . ,n

⇔ p^∗support. hyp. of H^∗=conv(h^∗₁, . . . ,h^∗_n) p^∗3h^∗₁, . . . ,h^∗_k

⇔ f^∗=conv(h^∗1, . . . ,h^∗k) is a (k−1)−face of H^∗

Duality between HandH^∗

The correspondence between the faces ofHandH^∗ isinvolutive and therefore bijective

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More generally and under the general position assumption, Letf be a(d−k)-face ofH and aff(f) =∩^ki=1h_i

p∈f ⇔ h^∗i ∈p^∗ for i=1, . . . ,k h^∗i ∈p^∗+ for i=k+1, . . . ,n

⇔ p^∗support. hyp. of H^∗=conv(h^∗₁, . . . ,h^∗_n) p^∗3h^∗₁, . . . ,h^∗_k

⇔ f^∗=conv(h^∗1, . . . ,h^∗k) is a (k−1)−face of H^∗

Duality between HandH^∗

The correspondence between the faces ofHandH^∗ isinvolutive and therefore bijective

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Algorithmic consequences

Depending on the application, the primal or the dual setting may be more appropriate

We will bound the combinatorial complexity of the intersection ofn upper half-spaces

We will compute the convex hull ofnpoints By duality, the results extend to the dual case

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Euler formula for 3-polytopes

The numbers of verticess, edgesaand facetsf of a polytope ofR³ satisfy

s−a+f =2 Schlegel diagram

s=s⁰ a⁰=a+ 1

f⁰=f+ 1 a⁰=a+ 1 f⁰=f s⁰=s+ 1

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Euler formula for 3-polytopes : s − a + f = 2

Incidences edges-facets

2a≥3f =⇒ a≤3s−6 f ≤2s−4 with equality when all facets are triangles

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Beyond the 3rd dimension

Upper bound theorem [McMullen 1970]

IfHis the intersection ofnhalf-spaces ofR^d nb faces of H= Θ(nb^d2c)

I anyk-face is the intersection ofd−khyperplanes definingH

I all vertices ofHare incident todedges and have distinctxd

the convex hull of < edges incident to a vertex

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Beyond the 3rd dimension

Upper bound theorem [McMullen 1970]

IfHis the intersection ofnhalf-spaces ofR^d nb faces of H= Θ(nb^d2c)

I anyk-face is the intersection ofd−khyperplanes definingH

I all vertices ofHare incident todedges and have distinctxd

the convex hull of < edges incident to a vertex

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Proof of the upper bound theorem

Bounding the number of vertices

1 ≥ d^d2eedges incident to a vertexpare inh⁺_p :xd≥xd(p)or inh⁻_p

⇒ pis axd-max orxd-min vertex of at least oned^d₂e-face ofH

⇒ # vertices ofH ≤2×#d^d₂e-faces ofH

2 Ak-face is the intersection ofd−khyperplanes definingH

⇒ #k-faces= n

d−k

=O(n^d−k)

⇒ #d^d₂e-faces=O(n^b^d²^c)

Bounding the total number of faces

The number of faces incident topdepends ondbut not onn

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Proof of the upper bound theorem

⇒ #k-faces= n

d−k

=O(n^d−k)

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Proof of the upper bound theorem

⇒ #k-faces= n

d−k

=O(n^d−k)

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Representation of a convex hull

Adjacency graph (AG) of the facets

In general position, all the facets are(d−1)-simplexes

Vertex

Face* v face Face

Vertex* vertex[d]

Face* neighbor[d]

i f

neighbor(ccw(i)) cw(i)

neighbor(i) ccw(i)

neighbor(cw(i))

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Incremental algorithm

Pⁱ : set of theipoints that have been inserted first

conv(Pⁱ): convex hull at stepi ^O

pi

conv(Ei) e s

t

f = [p1, ...,p_d]is aredfacet iff its supporting hyperplane separatesp_i fromconv(Pⁱ)

⇐⇒orient(p₁, ...,p_d,p_i)×orient(p₁, ...,p_d,O)<0

orient(p ,p , ...,p ) =

1 1 ... 1 =

1 1 ... 1

x01 x11 ... xd1

.. .. ..

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Update of conv( P

ⁱ

)

redfacet = facet whose supporting hyperplane separates oandp_i+1

horizon:(d−2)-faces shared by a blue and a red facet Updateconv(Pⁱ):

1 find the red facets

2 remove them and create the new facets

[pi+1,g],∀g∈horizon

O

pi

conv(Ei) e s

t

Complexity

proportional to the number of red facets

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Update of conv( P

ⁱ

)

redfacet = facet whose supporting hyperplane separates oandp_i+1

horizon:(d−2)-faces shared by a blue and a red facet Updateconv(Pⁱ):

1 find the red facets

2 remove them and create the new facets

[pi+1,g],∀g∈horizon

O

pi

conv(Ei) e s

t

Complexity

proportional to the number of red facets

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Complexity analysis

updateproportional to the number of red facets

# new facets=|conv(i,d−1)|

=O(i^b^d−1² ^c)

fast locate :insert the points in lexicographic order and search a 1st red facet instar(pi−1) (which necessarily exists)

O

pi

conv(Ei) e s

t

T(n,d) =O(nlogn) +Pn

i=1i^b^d−1² ^c)

=O(nlogn+nb^d+1² c)

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Complexity analysis

updateproportional to the number of red facets

# new facets=|conv(i,d−1)|

=O(i^b^d−1² ^c)

fast locate :insert the points in lexicographic order and search a 1st red facet instar(pi−1) (which necessarily exists)

O

pi

conv(Ei) e s

t

T(n,d) =O(nlogn) +Pn

i=1i^b^d−1² ^c)

=O(nlogn+nb^d+1² c)

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Lower bound

xi pi= (xi, x²i)

y=x² conv({pi}) =⇒tri({xi})

the orientation test reduces to 3 comparisons

orient(pi,pj,pk) =

xi−xj xi−xk

x²i −x²j x²i −x²k

= (xi−xj)(xj−xk)(xk−xi)

=⇒Lower bound :Ω(nlogn)

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Lower bound for the incremental algorithm

No incremental algorithm can compute the convex hull ofnpoints ofR³ in less thanΩ(n²)

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Randomized incremental algorithm

o: a point insideconv(P)

Pi : the set of the firstiinserted points

conv(Pi): convex hull at stepi ^O

pi

conv(Ei) e

Conflict graph

bipartite graph{pj} × {facets ofconv(Pⁱ)}

p_j†f ⇐⇒ j>i (pj not yet inserted) ∧ f∩op_j 6=∅

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Randomized incremental algorithm

o: a point insideconv(P)

Pi : the set of the firstiinserted points

conv(Pi): convex hull at stepi ^O

pi

conv(Ei) e

Conflict graph

bipartite graph{pj} × {facets ofconv(Pⁱ)}

p_j†f ⇐⇒ j>i (pj not yet inserted) ∧ f∩op_j 6=∅

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Randomized analysis

Hyp. : points are inserted in random order NotationsR: random sample of sizer ofP F(R) ={subsets ofdpoints ofR}

F₀(R) ={elements ofF(R)with0conflict inR}

(i.e.∈ conv(R)) F₁(R) ={elements ofF(R)with1conflict inR}

Ci(r,P) =E(|Fi(R)|)

(expectation over all random samplesR⊂ P of sizer)

Lemma

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Randomized analysis

Ci(r,P) =E(|Fi(R)|)

Lemma

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Randomized analysis

Ci(r,P) =E(|Fi(R)|)

Lemma

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Proof of the lemma : C

1

(r, P ) = C

0

(r, P ) = O(r b

^d²

c )

R⁰ =R\ {p}

f ∈F₀(R⁰)if f ∈F₁(R)andp†f (proba= ¹_r) or f ∈F₀(R)andR⁰ 3thedvertices off (proba= ^r−d_r ) Taking the expectation,

C₀(r−1,R) = 1

r |F₁(R)|+ r−d

r |F₀(R)| C₀(r−1,P) = 1

r C₁(r,P) +r−d

r C₀(r,P)

C₁(r,P) = d C₀(r,P)−r (C0(r,P)−C₀(r−1,P))

≤ d C₀(r,P)

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Randomized analysis 1

Updating the convex hull + memory space

Expected numberN(i)of facets created at stepi N(i) = X

f∈F(P)

proba(f ∈F₀(Pⁱ))×d i

= d i O

ib^d2c

= O(nb^d2c⁻¹)

Expected total number of created facets =O(nb^d2c)

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Randomized analysis 1

Updating the convex hull + memory space

Expected numberN(i)of facets created at stepi N(i) = X

f∈F(P)

proba(f ∈F₀(Pⁱ))×d i

= d i O

ib^d2c

= O(nb^d2c⁻¹)

Expected total number of created facets =O(nb^d2c)

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Randomized analysis2

Updating the conflict graph

Cost proportional to the number of faces ofconv(Pⁱ)in conflict withp_i+1 and somepj,j>i

N(i,j) =expected number of faces ofconv(Pⁱ)in conflict withpi+1 andpj,j>i Pi⁺=Pⁱ∪ {pi+1} ∪ {pj}: a random subset ofi+2points ofP

N(i,j) = X

f∈F(P)

proba(f ∈F2(Pi⁺))× 2

(i+1)(i+2)= 2C2(i+1)

(i+1)(i+2) =O(ib^d²c⁻²)

Expected total cost of updating the conflict graph

n

X

n

X N(i,j) =

n

X(n−i)O(ib^d²c⁻²) =O(nlogn+nb^d²c)

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Theorem

The convex hull ofnpoints ofR^d can be computed in time O(nlogn+nb^d2c)usingO(nb^d2c)space

The same bounds hold for computing theintersection ofn half-spacesofR^d

The randomized algorithm can bederandomized

[Chazelle 1992]

The same results hold forVoronoi diagramsprovided that d→d+1

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Voronoi diagram and Delaunay triangulation

Finite set of pointsP∈R^d

The Delaunay complex is the nerve of the Voronoi diagram

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Empty circumballs

An (open)d-ballBcircumscribing a simplexσ⊂ P is calledemptyif

1 vert(σ)⊂∂B

2 B∩ P =∅

Del(P)is the collection of simplices admitting an empty circumball

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Point sets in general position wrt spheres

P={p₁,p₂. . .p_n}is said to be in general position wrt spheres if 6 ∃d+2points ofPlying on a same(d−1)-sphere

Theorem [Delaunay 1936]

IfPis in general position wrt spheres, the natural mapping f :vert(DelP)→P

provides a realization ofDel(P)called the Delaunay triangulation ofP.

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Point sets in general position wrt spheres

P={p₁,p₂. . .p_n}is said to be in general position wrt spheres if 6 ∃d+2points ofPlying on a same(d−1)-sphere

Theorem [Delaunay 1936]

IfPis in general position wrt spheres, the natural mapping f :vert(DelP)→P

provides a realization ofDel(P)called the Delaunay triangulation ofP.

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Proof of Delaunay’s theorem 1

σ h(σ)

P

Linearization

S(x) =x²−2c·x+s, s=c²−r²

S(x)<0⇔

z<2c·x−s (h⁻_S )

z=x² (P)

⇔ˆx= (x,x²)∈h⁻_S

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Proof of Delaunay’s theorem 2

Proof of Delaunay’s th.

Pgeneral position wrt spheres

⇔Pˆ in general position σa simplex,Sσits circumscribing sphere

σ∈Del(P)⇔Sσ empty

⇔ ∀i, ˆpi∈h⁺_S_σ

⇔σˆis a face ofconv⁻(ˆP)

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Proof of Delaunay’s theorem 2

Proof of Delaunay’s th.

Pgeneral position wrt spheres

⇔Pˆ in general position σa simplex,Sσits circumscribing sphere

σ∈Del(P)⇔Sσ empty

⇔ ∀i, ˆpi∈h⁺_S_σ

⇔σˆis a face ofconv⁻(ˆP)

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Duality

V(P) =∩ih⁺_p

i D(P) =conv⁻(ˆP)

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Voronoi diagrams, Delaunay triangulations and polytopes

IfPis in general position wrt spheres :

V(P) =h⁺_p

1∩. . .∩h⁺_p

n

duality

−→ D(P) =conv⁻({ˆp₁, . . . ,ˆpn})

↑ ↓

Voronoi DiagramofP ^nerve−→ Delaunay ComplexofP

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Combinatorial complexity

The combinatorial complexity of the Delaunay triangulation diagram of npoints ofR^d is the same as the combinatorial complexity of a convex hull ofnpoints ofR^d+1

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Constructing Del(P), P = { p

1

, ..., p

_n

} ⊂ R

^d

Algorithm

1 Lift the points ofPonto the paraboloidxd+1=x²ofR^d+1: pi→ˆpi= (p_i,p²_i)

2 Computeconv({ˆpi})

3 Project the lower hullconv⁻({ˆpi})ontoR^d

Complexity :Θ(nlogn+n^b^d+1² ^c)

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Direct algorithm : insertion of a new point p

_i

1. Location : find all thed-simplices that conflict withp_i i.e. whose circumscribing ball containspi

2. Update :construct the newd-simplices

T

p_i

p_i pi

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Updating the adjacency graph

We look at thed-simplices to be removed and at their neighbors

Eachd-simplex is considered≤ ^d(d+1)₂ times

Update cost=O(# created and deleted simplices)

=O(# created simplices)

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Exercise : computing the DT of an ε-net

DefinitionLetΩbe a bounded subset ofR^dandPa finite point set inΩ.Pis called an(ε, η)-net ofΩif

1 Covering: ∀p∈Ω, ∃p∈P, kp−xk ≤ε

2 Packing: ∀p,q∈P,kp−qk ≥η

Questions

1 Show that(ε, ε)-nets exist

2 Show that any simplex with all its vertices at distance> εfrom∂Ω has a circumradius≤ε

3 Show that the complexity ofDel(P)isO(n)for fixedd

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Exercise : computing the DT of an ε-net

DefinitionLetΩbe a bounded subset ofR^dandPa finite point set inΩ.Pis called an(ε, η)-net ofΩif

1 Covering: ∀p∈Ω, ∃p∈P, kp−xk ≤ε

2 Packing: ∀p,q∈P,kp−qk ≥η

Questions

1 Show that(ε, ε)-nets exist

2 Show that any simplex with all its vertices at distance> εfrom∂Ω has a circumradius≤ε

3 Show that the complexity ofDel(P)isO(n)for fixedd