Thickness and Relaxation

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Thickness and Relaxation

Jean-Daniel Boissonnat

MPRI, Lecture 4

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Some optimality properties of Delaunay triangulations

Among all possible triangulations ofP,Del(P)

1 (2d) maximizes the smallest angle [Lawson]

2 (2d) Linear interpolation of{(pi,f(pi))}that minimizes [Rippa]

R(T) =P

i

R

T_i

∂φi

∂x

2

+

∂φi

∂y

2

dx dy (Dirichlet energy)

φi=linear interpolation of thef(pj)over triangleTi∈T

3 minimizes the radius of the maximal smallest ball

enclosing a simplex ) [Rajan]

ct=c⁰_t ct c⁰t rt r⁰t

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Some optimality properties of Delaunay triangulations

R(T) =P

i

R

T_i

∂φi

∂x

2

+

∂φi

∂y

2

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Some optimality properties of Delaunay triangulations

R(T) =P

i

R

T_i

∂φi

∂x

2

+

∂φi

∂y

2

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Optimizing the angular vector (d = 2)

Angular vector of a triangulationT(P)

ang(T(P)) = (α₁, . . . , α_3t), α₁≤. . .≤α_3t

Optimality Any triangulation of a given point setP whose angular vector is maximal (for the lexicographic order) is a Delaunay triangulation ofP

Good for matrix conditioning in FE methods

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Local characterization of Delaunay complexes

q1 f q2

σ1

σ2

Pair of regular simplices

σ2(q1)≥0 and σ1(q2)≥0

⇔ˆc1∈h⁺_σ₂andˆc2∈h⁺_σ₁

TheoremA triangulationT(P)such that all pairs of simplexes are regular is a Delaunay triangulationDel(P)

ProofThe PL function whose graphGis obtained by lifting the triangles is locally convex and has a convex support

⇒ G=conv⁻( ˆQ) ⇒ T(Q) =Del(Q)

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Local characterization of Delaunay complexes

q1 f q2

σ1

σ2

σ2(q1)≥0 and σ1(q2)≥0

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Local characterization of Delaunay complexes

q1 f q2

σ1

σ2

σ2(q1)≥0 and σ1(q2)≥0

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Constructive proof using flips

a

b c

d

a

b d c

t3 t4

a4 c4

d4

a3

b3 d3

a1 t1

c1

b1 c2

d2 t2

b2

While∃a non regular pair(t3,t4) /*t3∪t4 is convex */

replace(t3,t4)by(t1,t2)

Regularize⇔improveang(T(P)) ang(t1,t2)≥ang(t3,t4)

a1=a3+a4,d2=d3+d4,

c1≥d3, b1≥d4, b2≥a4, c2≥a3

I The algorithm terminates since the number of triangulations ofP is finite andang(T(P))cannot decrease

I The obtained triangulation is a Delaunay triangulation ofPsince all its edges are regular

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Constructive proof using flips

a

b c

d

a

b d c

t3 t4

a4 c4

d4

a3

b3 d3

a1 t1

c1

b1 c2

d2 t2

b2

a1=a3+a4,d2=d3+d4,

c1≥d3, b1≥d4, b2≥a4, c2≥a3

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Constructive proof using flips

a

b c

d

a

b d c

t3 t4

a4 c4

d4

a3

b3 d3

a1 t1

c1

b1 c2

d2 t2

b2

a1=a3+a4,d2=d3+d4,

c1≥d3, b1≥d4, b2≥a4, c2≥a3

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Flat simplices may exist in higher dimensional DT

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0$"('12'3%,4$+/,'".$&#4$+/,'

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##

345$+6&*(+%7#%$8)06"#,0&6'"%#

####################5#()./0&(#9"&:""+#3#/0&"8*0;%

<45$+6&(+%7#=1*&0;#"=1"%

####################5#&"-0&(#9"&:""+#-#(8#/(8"#/0&"8*0;%

>45$+6&(+%7#6(8+"8#?"8&6"%

5#1,"-$0&(#9"&:""+#@#(8#/(8"#/0&"8*0;%#

Each square face can be circumscribed by an empty sphere This remains true if the grid points are slightly perturbed therefore creating flat tetrahedra

Computational Geometry Learning Thickness and relaxation MPRI, Lecture 4 6 / 35

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Simplex quality

Altitudes

D(q, σ)

σq

q Ifσq, the face oppositeqinσ is protected, Thealtitudeofqinσ is

D(q, σ) =d(q,aff(σq)),

whereσqis the face oppositeq.

Definition (Thickness [Cairns, Whitney, Whitehead et al.]) Thethicknessof aj-simplexσ with diameter∆(σ)is

Θ(σ) =

(1 ifj=0 minp∈σ D(p,σ)

j∆(σ) otherwise.

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Simplex quality

Altitudes

D(q, σ)

σq

q Ifσq, the face oppositeqinσ is protected, Thealtitudeofqinσ is

D(q, σ) =d(q,aff(σq)),

whereσqis the face oppositeq.

Definition (Thickness [Cairns, Whitney, Whitehead et al.]) Thethicknessof aj-simplexσ with diameter∆(σ)is

Θ(σ) =

(1 ifj=0 minp∈σ D(p,σ)

j∆(σ) otherwise.

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Protection

δ

cσ

δ-protectionWe say that a Delaunay simplexσ⊂Lisδ-protected if kc_σ−qk>kc_σ−pk+δ ∀p∈σ and ∀q∈L\σ.

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Protection implies thickness

If P is a(ε,η)-net, with¯ δ= ¯δε,

∀x∈Ω, d(x,P)≤ε

∀p,q∈P, kp−qk ≥ηε¯

then the thickness of all Delaunay simplices is lower bounded Θ(σ)>Θ0= ^¯^δ^(¯^η+¯_8d^δ)

(except possibly near the boundary ofconv(P))

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Protection implies thickness (Proof 1)

c c⁰ H

δ q q^∗

candc⁰are on the same side ofH

σ d-simplex ofDel(P)

σ =q∗τ σ⁰⊃τ ∧ q6∈σ⁰ H=aff(τ)

q^∗ ∈B⁰ ⇒ kqq^∗k ≤δ

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Protection implies thickness (Proof)

c⁰ c

a

b q H δ

S⁰ r S

q^∗

Hseparatescandc⁰

σd-simplex ofDel(P)

σ=q∗τ σ⁰⊃τ ∧ q6∈σ⁰ H=aff(τ)

γ =∠qab, α=∠qac, β =∠cab

wlogγ ≥∠qba

γ =α+β≥ ^π₂ (otherwise easy)

cosα= ^ka−qk_2r ≥ _2ε^δ cosβ = ^ka−bk_2r ≥ _2ε^η

kqq^∗k=kaqk sinγ > δsinγ

> _4ε^δ (η+δ)

∆(σ)≤2ε

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The Lov ´asz Local Lemma

Motivation

Given: A set of (bad) eventsA₁, ...,AN, each happens withproba(Ai)≤p<1

Question :what is the probability that none of the events occur?

The case of independent events

proba(¬A₁∧...∧ ¬A_N)≥(1−p)^N >0

What if we allow a limited amount of dependency among the events?

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The Lov ´asz Local Lemma

Motivation

proba(¬A₁∧...∧ ¬A_N)≥(1−p)^N >0

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The Lov ´asz Local Lemma

Motivation

proba(¬A₁∧...∧ ¬A_N)≥(1−p)^N >0

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LLL : symmetric version [Lov ´asz & Erd ¨os 1975]

Under the assumptions

1 proba(Ai)≤p<1

2 A_i depends of≤Γother eventsA_j

3 proba(Ai)≤ e(Γ+1)¹ e=2.718...

then

proba(¬A₁∧...∧ ¬A_N)>0

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Moser and Tardos’ constructive proof of the LLL [2010]

P a finite set of mutually independent random variables

Aa finite set of events that are determined by the values ofS⊆ P Two events are independent iff they share no variable

Algorithm

for allP∈ P do

v_P←a random evaluation ofP;

while∃A∈ A:Ahappens when (P=v_P,P∈ P)do

pick an arbitrary happening eventA∈ A; for allP∈variables(A)do

vP←a new random evaluation ofP;

return(vP)P∈P;

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Moser and Tardos’ constructive proof of the LLL [2010]

P a finite set of mutually independent random variables

Aa finite set of events that are determined by the values ofS⊆ P Two events are independent iff they share no variable

Algorithm

for allP∈ P do

v_P←a random evaluation ofP;

while∃A∈ A:Ahappens when (P=v_P,P∈ P)do pick an arbitrary happening eventA∈ A;

for allP∈variables(A)do

vP←a new random evaluation ofP;

return(vP)P∈P;

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Moser and Tardos’ theorem

if

1 proba(Ai)≤p<1

2 Ai depends of≤Γother eventsAj

3 proba(Ai)≤ e(Γ+1)¹ e=2.718...

then∃an assignment of values to the variablesP such that no event inAhappens

The randomized algorithm resamples an eventA∈ Aat most _Γ¹ expected times before it finds such an evaluation

The expected total number of resampling steps is at most ^N_Γ

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Home work

Read the beautiful (rather simple) proof of Moser & Tardos Learn about the parallel and the derandomized versions Listen to a talk by Aravind Srinivasan on further extensions

https://video.ias.edu/csdm/2014/0407-AravindSrinivasan

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Protecting Delaunay simplices via perturbation

Picking regions: pickp⁰ inB(p, ρ)

Sampling parameters of a perturbed point set If P is an(ε,η)-net,¯ P⁰ is an(ε⁰,η¯⁰)-net, where

ε⁰ =ε(1+ ¯ρ) and η¯⁰ = η¯−2ρ¯ 1+ ¯ρ

Notation:¯x= _ε^x

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Protecting Delaunay simplices via perturbation

Picking regions: pickp⁰ inB(p, ρ)

Sampling parameters of a perturbed point set

If P is an(ε,η)-net,¯ P⁰ is an(ε⁰,η¯⁰)-net, where ε⁰ =ε(1+ ¯ρ) and η¯⁰ = η¯−2ρ¯

1+ ¯ρ

Notation:¯x= _ε^x

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The LLL framework

Random variables : P⁰ a set of random points{p⁰, p⁰ ∈B(p, ρ),p∈P} Event: ∃φ⁰ = (σ⁰,p⁰) (Bad configuration)

σ⁰ is adsimplex withR_σ⁰ ≤ε+ρ

p⁰ ∈Z_δ(σ⁰) Z_δ(σ⁰) =B(c_σ⁰,R_σ⁰+δ)\B(c_σ⁰,R_σ⁰)

Algorithm Input: P,ρ,δ

whilean eventφ⁰ = (σ⁰,p⁰)occursdo resample the points ofφ⁰

updateDel(P⁰)

Output: P⁰ andDel(P⁰)

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The LLL framework

Random variables : P⁰ a set of random points{p⁰, p⁰ ∈B(p, ρ),p∈P} Event: ∃φ⁰ = (σ⁰,p⁰) (Bad configuration)

σ⁰ is adsimplex withR_σ⁰ ≤ε+ρ

p⁰ ∈Z_δ(σ⁰) Z_δ(σ⁰) =B(c_σ⁰,R_σ⁰+δ)\B(c_σ⁰,R_σ⁰)

Algorithm Input: P,ρ,δ

whilean eventφ⁰ = (σ⁰,p⁰)occursdo resample the points ofφ⁰

updateDel(P⁰)

Output: P⁰ andDel(P⁰)

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Bounding Γ

Lemma : An event is independent of all but at mostΓother bad events whereΓdepends onη,¯ ρ,¯ ¯δand exponentially ond

Proof:

Letφ⁰= (σ⁰,p⁰)be a bad configuration.

∀p⁰ ∈φ⁰, kp⁰−cσ⁰k ≤Rσ⁰+δ=R=ε+ρ+δ≤ε(1+ ¯ρ+ ¯δ) two configurations(σ⁰,p⁰)and(τ⁰,q⁰)such thatkcσ⁰−cτ⁰k>2Rcannot have a vertex in common and therefore the corresponding events are independent

the number of events that may not be independent from an event(σ⁰,p⁰) is at most the number of(d+1)-simplices with vertices inB(cσ⁰,3R).

Since P⁰isη⁰-sparse,

Γ = 3R+^η₂⁰

η⁰

!^d+2

=

1+6(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

^d+2

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Bounding Γ

Proof:

Γ = 3R+^η₂⁰

η⁰ 2

!^d+2

=

1+6(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

^d+2

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Bounding Γ

Proof:

Γ = 3R+^η₂⁰

η⁰

!^d+2

=

1+6(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

^d+2

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Bounding Γ

Proof:

Γ = 3R+^η₂⁰

η⁰ 2

!^d+2

=

1+6(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

^d+2

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Bounding Γ

Proof:

Γ = 3R+^η₂⁰

η⁰

!^d+2

=

1+6(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

^d+2

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Bounding proba(A

i

)

p

B(p, ρ)

2β S(c, R)

S1

θ R ρ

c

S(c,R)a hypersphere ofR^d

Tδ =B(c,R+δ)\B(c,R) Bρanyd-ball of radiusρ <R

vold(Tδ∩Bρ)≤Ud−1 π 2ρd−1

δ,

proba(p⁰ ∈Z_δ(σ⁰))≤$= _π² ^δ_ρ

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Bounding proba(A

i

)

p

B(p, ρ)

2β S(c, R)

S1

θ R ρ

c

S(c,R)a hypersphere ofR^d

Tδ =B(c,R+δ)\B(c,R) Bρanyd-ball of radiusρ <R

vold(Tδ∩Bρ)≤Ud−1 π 2ρd−1

δ,

proba(p⁰ ∈Z_δ(σ⁰))≤$= _π² ^δ_ρ

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Bound on the number of events

Σ(p⁰): number ofd-simplices that can possibly make a bad configuration withp⁰ ∈P⁰ for some perturbed setP⁰

R=ε+ρ+δ

X

p⁰∈P⁰

Σ(p⁰)≤ |P⁰ ∩ B(p⁰,2R)| ≤N=n

1+4(1+ ¯ρ+ ¯δ) (1+ ¯ρ)

¯ η−2¯ρ

d+1

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Main result

Under condition

2e

π (Γ +1)δ ≤ρ < η 2 the algorithm terminates.

Guarantees on the output

I d_H(P,P⁰)≤ρ

I thed-simplices ofDel(P⁰)are _{1+ ¯}^δ_ρ-protected

I and therefore have a positive thickness

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Main result

Under condition

2e

π (Γ +1)δ ≤ρ < η 2 the algorithm terminates.

Guarantees on the output

I d_H(P,P⁰)≤ρ

I thed-simplices ofDel(P⁰)are _{1+ ¯}^δ_ρ-protected

I and therefore have a positive thickness

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Complexity of the algorithm

The number of resamplings executed by the algorithm is at most N/Γ≤C n

whereCdepends onη,¯ ρ,¯ δ¯andd

Each resampling consists in perturbingO(1)points

Updating the Delaunay triangulation after each resampling takes O(1)time

The expected complexity islinear in the number of points

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Witness Complex

^{[de Silva]}

La finite set of points (landmarks) inR^d vertices of the complex

W a dense sample (witnesses) pseudo circumcenters

The Witness Complex

Wit(L, W)

w

Definition

2Wit(L, W) () 8⌧ 9w2W with

d(w, p)d(w, q) 8p2 8q2L\

L:Landmarks (black dots); the vertices of the complex W :Witnesses (blue dots)

R. Dyer (INRIA) Del(L,M) =Wit(L, W) Assisi, EuroCG 2012 2 / 10

Letσbe a (abstract) simplex with vertices inL, and letw∈W. We say thatwis awitnessofσif

kw−pk ≤ kw−qk ∀p∈σ and ∀q∈L\σ.

Thewitness complexWit(L,W)is the complex consisting of all simplexes

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Witness Complex

^{[de Silva]}

The Witness Complex

Wit(L, W)

w

Definition

d(w, p)d(w, q) 8p2 8q2L\

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Witness Complex

^{[de Silva]}

The Witness Complex

Wit(L, W)

w

Definition

d(w, p)d(w, q) 8p2 8q2L\

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Easy consequences of the definition

The witness complex can be defined in any metric space and, in particular, for discrete metric spaces

IfW⁰⊆W, thenWit(L,W⁰)⊆Wit(L,W) Del(L)⊆Wit(L,R^d)

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Easy consequences of the definition

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Easy consequences of the definition

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First identity theorem

[de Silva 2008]

Theorem : Wit(L,W)⊆Wit(L,R^d) =Del(L) Remarks

I Faces of all dimensions have to be witnessed

I Wit(L,W)isembeddedinR^d ifLis in general position wrt spheres

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First identity theorem

[de Silva 2008]

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First identity theorem

[de Silva 2008]

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Proof of de Silva’s theorem

τ = [p0, ...,pk]is ak-simplex ofWit(L)witnessed by a ballBτ (i.e.Bτ ∩L=τ) We prove thatτ∈Del(L)by a double induction on

k

l=|∂Bτ∩τ|

Clearly true fork=0and|∂Bτ∩τ|=1

Bσ

c

w

Bτ

τ σ

Hyp.: true for k⁰≤k−1 and l≤k σ=∂Bτ∩τ

σ∈Del(L)by the hyp.

Scentered on[cw],σ⊂S,|S∩τ|=l+1 Switnessesτ

proceed by induction untill=k+1,

⇒τ∈Del(L)

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Proof of de Silva’s theorem

τ = [p0, ...,pk]is ak-simplex ofWit(L)witnessed by a ballBτ (i.e.Bτ ∩L=τ) We prove thatτ∈Del(L)by a double induction on

k

l=|∂Bτ∩τ|

Clearly true fork=0and|∂Bτ∩τ|=1

Bσ

c

w

Bτ

τ σ

Hyp.: true for k⁰≤k−1 and l≤k σ=∂Bτ∩τ

σ∈Del(L)by the hyp.

Scentered on[cw],σ⊂S,|S∩τ|=l+1 Switnessesτ

proceed by induction untill=k+1,

⇒τ∈Del(L)

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Case of sampled domains : Wit(L, W) 6 = Del(L)

W a finite set of points⊂Ω

Wit(L,W)6=Del(L), even ifWis a dense sample ofΩ

a b

Vor(a, b)

[ab]∈Wit(L,W) ⇔ ∃p∈W, Vor2(a,b)∩W 6=∅

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Protection implies Wit(L, W) = Del(L)

LemmaIf ad-simplexσ ofDel(L)isδ-protected withδ ≥2ε, thenσ ∈Wit(L,W)

If true for alld-simplices ofDel(L), thenWit(L,W) =Del(L).

Proof

1 kcσ−pik=kcσ−pjk=r ∀pi,pj∈σ

2 kcσ−plk>r+δ⁰ ∀pl∈L\σ

∀x∈B(cσ, δ⁰/2),∀pi ∈σand∀pl∈L\σ,

|x−pi| ≤ |cσ−pi|+|cσ−x| ≤r+σ δ⁰2

and |x−pl| ≥ |cσ−pl| − |x−cσ|>r+δ⁰−^δ₂⁰ =r+^δ₂⁰

Hence,xis a witness ofσ. Ifε≤δ⁰/2, there must be a pointw∈WinB(c, δ⁰/2) which witnessesσ.

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Protection implies Wit(L, W) = Del(L)

LemmaIf ad-simplexσ ofDel(L)isδ-protected withδ ≥2ε, thenσ ∈Wit(L,W)

If true for alld-simplices ofDel(L), thenWit(L,W) =Del(L).

Proof

1 kcσ−pik=kcσ−pjk=r ∀pi,pj∈σ

2 kcσ−plk>r+δ⁰ ∀pl∈L\σ

∀x∈B(cσ, δ⁰/2),∀pi ∈σand∀pl∈L\σ,

|x−pi| ≤ |cσ−pi|+|cσ−x| ≤r+σ δ⁰2

and |x−pl| ≥ |cσ−pl| − |x−cσ|>r+δ⁰−^δ₂⁰ =r+^δ₂⁰

Hence,xis a witness ofσ. Ifε≤δ⁰/2, there must be a pointw∈WinB(c, δ⁰/2) which witnessesσ.

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Pseudomanifolds

A simplicial complexK is ak-pseudomanifold complexif

1 Kis a purek-complex

2 every(k−1)-simplex is the face of exactly twok-simplices

3 the adjacency graph of thek-simplices is connected

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Pseudomanifolds

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Pseudomanifolds

Lemma

LetK be a triangulation of P andK⁰ a simplicial complex with the same vertex set

IfK⁰⊆K and K⁰ is a pseudomanifold, then K⁰ =K.

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Good links implies Wit(L, W ) = Del(L)

Lemma

IfWis anε-sample andWit(L,W)is a pseudomanifold complex, then Wit(L,W) =Del(L).

Proof

I Wit(L,W)⊆Del(L)

I Del(L)contains a triangulation

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Good links implies Wit(L, W ) = Del(L)

Lemma

IfWis anε-sample andWit(L,W)is a pseudomanifold complex, then Wit(L,W) =Del(L).

Proof

I Wit(L,W)⊆Del(L)

I Del(L)contains a triangulation

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Construction of witness complexes

Input: L,W,kp−wkfor allp∈L,w∈W Wit(L,W) :=∅

W⁰:=W

foreachw∈W⁰do

compute the listN(w) = (p₀(w), ...,p_|L|−1(w))of the points ofL sorted by distance fromw

fori=0, ...,|L| −1do foreachw∈W⁰ do

ifthei-simplexσ(w) = (p0(w),p₂(w), ...,pi(w))6∈Wit(L,W)then ifthe(i−1)-faces ofσ(w)are all inWit(L,W)then

addσ(w)toWit(L,W) else

W⁰ :=W⁰\ {w} Output: Wit(L,W)

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Complexity

firstforloopO(|W| × |L|log|L|)

I can be improved toO(|W|)isLis a net in general position wrt

spheres (Exercise)

secondforloop(|Wit(L,W)|+|W|) log|L|

I at each step, either we create a new simplex or we remove a witness fromW⁰

I a simplex can be searched in timeO(log|Wit(L,W)|=O(log|L|)

Thickness and Relaxation