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Points, Distances, and Cellular Automata: Geometric and Spatial - - PowerPoint PPT Presentation

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Luidnel Maignan Alchemy : INRIA Saclay , LRI , Univ. Paris XI luidnel.maignan@inria.fr New

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Luidnel Maignan Alchemy: INRIA Saclay, LRI, Univ. Paris XI luidnel.maignan@inria.fr New World of Computation 2011 Orl´ ean - 23-24 May 2011

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Introduction

Spatial Computing and Cellular Automata Massively Distributed Systems ⇒ Spatial Features

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Introduction

Spatial Computing and Cellular Automata Massively Distributed Systems ⇒ Spatial Features Why? Physics and Locality

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Introduction

Spatial Computing and Cellular Automata Massively Distributed Systems ⇒ Spatial Features Why? Physics and Locality Exemple? Computer Architecture, Communication

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Introduction

Spatial Computing and Cellular Automata Spatial Computing: focus on space Cellular Automata: simple framework, precise results

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Global Statement

In the same manner that geometry is deeply based on distances, basing spatial algorithmics on the intrinsic metric of the spatial computers leads to more precise and generic formulation.

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics

Outline

1 Space, Time, and Cellular Automata 2 Distance Fields and Gradients 3 Density Uniformisation 4 Convex Hulls 5 Gabriel graphs 6 Conclusion and Perpectives

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Space, Time, and Cellular Automata

Space, Time, and Cellular Automata

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Space, Time, and Cellular Automata Classical Considerations

Cellular Automata

Cellular Automata Regular lattice of cells, also called sites, (or points) All sites states are updated synchronously State updates depends only on neighborhood sites states 4-Square 8-Square Hexagonal

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Space, Time, and Cellular Automata Classical Considerations

Cellular Automata

Cellular Automata Regular lattice of cells, also called sites, (or points) All sites states are updated synchronously State updates depends only on neighborhood sites states 4-Square 8-Square Hexagonal

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Space, Time, and Cellular Automata Classical Considerations

Cellular Automata

Cellular Automata Regular lattice of cells, also called sites, (or points) All sites states are updated synchronously State updates depends only on neighborhood sites states 4-Square 8-Square Hexagonal

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Space, Time, and Cellular Automata Classical Considerations

Cellular Automata and Distances

Directions and Distances Traditionnaly, neighbors are named North, South, East, West In this work, no direction, only the graph and its metric Distances only ⇔ Rotational invariance 4-Square 8-Square Hexagonal

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients

Distance Fields and Gradients

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Classical Definition and Computation

Definition (Distance Map) The distance map DP of a given set of particles P associates to each point x its distance d(x, y) to the closest particle y ∈ P. DP(x) = d(P, x) = min{ d(x, y) | y ∈ P }. Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

2 2 2 2 1 0 1 2 2 2 2 1 0 1 2 2 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

3 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

4 4 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 2 0 2 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 1 0 1 3 3 3

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 2 1 0 1 2 1 0 1 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 3 2 1 0 1 2 3 2 1 0 1 2 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Classical Definition and Computation

Distance Field Evolution

Classical Distance Field D[P]t+1(x) =

  • 0 if x ∈ Pt else:

min{ 1 + D[P]t(y) | y ∈ N(x) }.

4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

2 2 2 2 1 0 1 2 2 2 2 1 0 1 2 2 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

3 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

4 4 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 2 1 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 2 0 0

.5 1 2 3 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 3 1 0 1 1

.5

3 3 3

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 4 2 1 0 1 2 2

.5

1 0 1 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

5 3 2 1 0 1 2 3 3

.5

2 1 0 1 2 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients Corrected Distance Field Evolution

Corrected Distance Field Evolution

Corrected Distance Field D[P]t+1(x) =      0 if x ∈ Pt+1 else: 0.5 if x ∈ Pt else: min{ 1 + D[P]t(y) | y ∈ N(x) }.

4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 2 1 0 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field

Checkpoint We have: distances locally, globally, and dynamically We don’t have: finite number of states

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field

Checkpoint We have: distances locally, globally, and dynamically We don’t have: finite number of states Bounded information No bound on distances Bounded gradient (differences between neighboring sites) What about modulo ?

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

2 2 2 2 0 0

.5 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 0 .5 0 2 2 2 2 2 2

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

3 3 3

.5 2 3 3 3 3 3 3 3 3 3 3 3 3 2 .5 .5

3 3 3 3 3

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

4 4

☎ 1 ☎ 3 4 4 4 4 4 4 4 4 4 4 3 ☎
  • ☎
  • ☎
  • 4 4 4 4

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

5

.5 0 1 .5 2 3 3 .5 4 5 5 5 5 5 5 5 5 4 3 .5 2 1 .5 1 0 .5 0 2 .5 3 5 5 5

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

4

✁ ✂ ✁ ✂

2

✁ ✂ 3 4 4 ✁ ✂ 5 6 6 6 6 6 6 5 4

.5 3 2 .5 2 1 .5 1 0 1 3 .5 4 6 6

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient

3 2

.5 2 .5

3

.5 4 5 5 .5 6 7 7 7 7 6 .5

3

.5 3 2 .5 2 .5 1 2 .5 5 7

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient In this case: 2 consecutive moves ⇒ gradient bound of 3

3

.5 3 .5 .5 1 .5 5 6 6 .5 7 8 8 7 6 .5 5 4 .5 4 3 .5 3 1 0 .5 0 1 .5 2 3 5 .5 6

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

From Infinite To Finite Field (Cont.)

Modulo in action Particles maximal speed determines maximal gradient In this case: gradient bound of 3 ⇒ modulo 7

3

.5 3 .5 .5 1 .5 5 6 .5 .5 .5 0 .5

5

.5

3

.5 3 .5 .5 2 3 5 .5 6

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Distance Fields and Gradients From Infinite To Finite Field

Building on top of distances

Distance fields as building blocks Moving according to the distance field Detecting patterns of distances and particles Case Study Density Uniformisation (unidimensional) Convex Hull (multidimensional) Gabriel Graph (multidimensional)

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation

Density Uniformisation

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Statement

Problem Statement

Problem Definition Move the particles to a uniform distribution Input: Output:

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Analysis

Problem Analysis

Intuition Each particle needs to occupy its space Boundary between individual spaces ⇔ middles Occupy its space ⇔ be at the middles

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Problem Analysis

Application: 1D Uniformisation

Solution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting system

Initial system state p0(x) = x ∈ P w0(x) = x ∈ P System fields composition        dp = D[p] dw = D[w] p = M[p0, B[dp] ∧ Dir[dw, ≤]] w = M[w0, B[dw] ∧ Dir[dp, ≤]]

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 98

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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slide-99
SLIDE 99

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 100

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 101

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 102

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 103

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 104

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 105

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 106

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 107

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

The resulting evolution

Signal and Dynamics We can see that signals travels through the space We can assign energy and momentum to these signals Defined by fields; composed for global system

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SLIDE 108

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Density Uniformisation Solution Description

Space-time diagram of the uniformisation

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SLIDE 109

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls

Convex Hulls

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SLIDE 110

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Euclidean Space

Definition (Euclidean convex region) A convex region contains all segments joining two of its points Convex Polygon Concave Polygon

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SLIDE 111

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Euclidean Space (Cont.)

Definition (Convex Hull) The convex hull is the smallest convex region containing a set

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SLIDE 112

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Euclidean Space (Cont.)

Definition (Convex Hull) The convex hull is the smallest convex region containing a set

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SLIDE 113

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Euclidean Space (Cont.)

Definition (Convex Hull) The convex hull is the smallest convex region containing a set

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SLIDE 114

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Cellular Space

Definition (Metric convex region) A convex region contain all shortest paths joining two of its points Convexity for Metric Cellular Space

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SLIDE 115

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Definition and Problem Statement

Convexity in Cellular Space (Cont.)

Shortest paths between two points Many shortest path between two points: Interval [x, y] = { z ∈ S | d(x, z) + d(z, y) = d(x, y) }

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SLIDE 116

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 117

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 118

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 119

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 120

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 121

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 122

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 123

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 124

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 125

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 126

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 127

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 128

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 129

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 130

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 131

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 132

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 133

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 134

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 135

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 136

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 137

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 138

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 139

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 140

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 141

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 142

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 143

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls First Step: Local convexity

First Step: Local convexity

Definition convt(x) = ∃y0, y1 ∈ {y ∈ N(x) | y ∈ Pt ∨convt−1(y)}; x ∈ [y0, y1]

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SLIDE 144

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles)

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SLIDE 145

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles)

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SLIDE 146

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles)

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SLIDE 147

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles)

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SLIDE 148

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles)

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SLIDE 149

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles) Detect the middles of the shortest paths

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SLIDE 150

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles) Detect the middles of the shortest paths Go back from the middles to the particles

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SLIDE 151

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles) Detect the middles of the shortest paths Go back from the middles to the particles

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SLIDE 152

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles) Detect the middles of the shortest paths Go back from the middles to the particles

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SLIDE 153

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Second Step: Global Convexity for Two Particles

Second Step: Global Convexity for Two Particles

Required Fields: Grow a distance field modulo 3 (static particles) Detect the middles of the shortest paths Go back from the middles to the particles

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SLIDE 154

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Last Step: Global Convexity for Many Particles

Last Step: Global Convexity for Many Particles

Convex Hull of Two Points Grow a distance field modulo 3 Detect the middles of the shortest paths Go back from the middles to the points

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SLIDE 155

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Last Step: Global Convexity for Many Particles

Last Step: Global Convexity for Many Particles

Convex Hull of Many Points Grow a distance field modulo 3 Detect the middles of the shortest paths Go back from the middles to the points

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SLIDE 156

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 157

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 158

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 159

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 160

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 161

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 162

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 163

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 164

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 165

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 166

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 167

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Real Challenge: Global Convexity

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SLIDE 168

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Proximity Graph Characterisation

Distance Field and Voronoi Diagram

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SLIDE 169

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Proximity Graph Characterisation

Delaunay graph ?

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SLIDE 170

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Proximity Graph Characterisation

Delaunay graph ? No, only a subset

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SLIDE 171

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Convex Hulls Solution Description: Global Convexity for Many Particles

Proximity Graph Characterisation

Gabriel Graph !

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SLIDE 172

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs

Gabriel graphs

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SLIDE 173

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Original Gabriel graphs

Original Gabriel graphs

Definition (Gabriel Graph) Euclidean spaces Connects two particles x and y if and only if the ball using the segment [xy] as diameter does not contain any other particle. Gabriel graphs on Cellular Spaces Connected for Euclidean... and for cellular spaces ?

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SLIDE 174

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Original Gabriel graphs

Gabriel graphs on Cellular Spaces

(a) Each point (b) Each line (c) Each diagonal (d) Subcase of (c) Generalisation Connectedness is not ensured in general The cause is the non-uniqness of diameters and minimal balls We need to generalize the definition

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SLIDE 175

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs

Principles of Gabriel Graphs

Connectedness, Minimality, and Locality Minimality and connectedness:

minimum spanning trees

Locality and connectedness :

arbitrarily choice implies global coherence union of all minimum spanning trees

Locality and minimality :

Edge decision should be local union of all local minimum spanning trees

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SLIDE 176

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs

From Principles to Definition

Definition (Metric Gabriel Graph) Any metric space Connects two particles x and y if and only if there is a ball B(c, r) such that d(x, y) = 2r and {x, y} is an edge of a minimum spanning tree of P ∩ B(c, r).

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SLIDE 177

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs

From Principles to Definition (Cont.)

Definition (Metric Gabriel Graph) Any metric space Connects two particles x and y of P if and only if there is a ball B(c, r) such that there is a cut {P0, P1} of P ∩ B(c, r) with (x, y) ∈ P0 × P1 and d(P0, P1) = 2r.

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SLIDE 178

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs

From Principles to Definition (Cont.)

Definition (Metric Gabriel Graph) Any metric space Connects two particles x and y of P if and only if there is a ball B(c, r) such that there is a cut {P0, P1} of P ∩ B(c, r) with (x, y) ∈ P0 × P1 and d(P0, P1) = 2r. Preservation of the Properties Metric Gabriel graphs are always connected On Euclidean spaces, they are Gabriel graphs

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SLIDE 179

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

Distance fields and dilations

Example of a metric Gabriel ball center

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SLIDE 180

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

Distance fields and dilations (Cont.)

Example of a non-metric Gabriel ball center

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SLIDE 181

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

Dilations and interval slices

One particle Two particles r = 4 r = 3

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SLIDE 182

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The metric Gabriel ball centers field

centt(x) =            ⊥ if t = 0 ⊤ if centt(x, x) ⊤ if ∃y ∈ N(x), centt(x, y) ⊥ otherwise; centt(x, y) = |Qt(x, y)/C +

2rxy | ≥ 2

Qt(x, y) = { z ∈ B(xy, rxy) | D[P]t−1(z) + rxy = D[P]t−1(x, y) }

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SLIDE 183

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 184

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 185

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 186

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 187

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 188

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 189

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 190

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 191

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Gabriel graphs Metric Gabriel Graphs on Cellular Automata

The resulting cellular automaton

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SLIDE 192

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Conclusion and Perpectives

Conclusion and Perpectives

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SLIDE 193

Points, Distances, and Cellular Automata: Geometric and Spatial Algorithmics Conclusion and Perpectives

Perspectives

In the same framework Voronoi Diagram Field Firing Squad Synchronisation Problem Extending the framework Cayley Graphs Asynchronicity Amorphous Computers

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