Comment reconstruire le graphe de visibilit

1
(No Transcript)
2
(No Transcript)
3
Visibility Graph of Polygons

• Gvis(P)
• Vertices of Gvis ltgt vertices of P
• Edge (u,v) in Gvis iff
• u and v can see each-other.
• (if the line u to v is inside P)

• Objective
• Construct Gvis
• (up to isomorphism)

4
(No Transcript)
5
(No Transcript)
6
Polygons vs. Graphs

• Exploration of Polygons Exploration of
  edge-labeled graphs

7
Polygons vs. Graphs II

• Exploration of Polygons Exploration of
  edge-labeled graphs

Property YK 1996 A robot exploring an
edge-labeled graph of known size n, can not
always reconstruct the graph.
8
Polygons vs. Graphs III

• Exploration of Polygons Exploration of
  edge-labeled graphs

Property YK 1996 A robot exploring an
edge-labeled graph of known size n, can not
always reconstruct the graph.
Theorem A robot exploring a polygon can always
reconstruct the visibility graph of the polygon
if it knows an upper bound on n.
9
Graph Exploration

• The view of an exploring robot

Minimum-base
10
Graph Exploration II

• The minimum-base of a graph G
• The smallest graph B such that G covers B

?
Property YK 1996 A robot exploring an
edge-labeled graph can construct the minimum-base
if it knows an upper bound on n.
11
Exploring visibility graphs

• If C1,C2,,Cp are the classes of vertices in Gvis
• Ci q n/p
• Classes repeat periodically on the boundary.
• How to find the internal edges (chords)?

12
Properties of Polygons

• Every polygon has an ear!
• ( If a,b,c appear in this order on the boundary
  b is an ear iff a sees c.)
• Removing an ear of a simple polygon leaves a
  smaller polygon.
• (visibility relationships are maintained)
• Every sub-polygon of four or more vertices has a
  chord.

13
Properties of Polygons II

• It is easy to recognize an ear!
• Check for the paths
• (1, -1) (-2, 2) (1, -1)
• (-1, 1) (2, -2) (-1, 1)

-1
-2
1
2
-1
1
Lemma If v is an ear, every vertex in the
class of v is an ear.
14
Deconstructing Polygons

• Choose a class Ci of ears.
• Remove all Ci vertices from P
• (i.e. remove a vertex from B)

15
Deconstructing Polygons-II

• Choose a class Ci of ears.
• Remove all Ci vertices from P
• (i.e. remove a vertex from B)

Repeat until a single class remains!
16
Deconstructing Polygons III

• Choose a class Ci of ears.
• Remove all Ci vertices from P
• (i.e. remove a vertex from B)

Repeat until a single class remains!
Remaining vertices form a clique!
Lemma There is a unique class C which forms a
clique.
17
Using class C
Lemma There is a unique class C which forms a
clique.

• C corresponds to a vertex with q-1 self loops in
  the minimum-base.
• gt
• Robot can compute n pq

18
Solving Rendezvous
Rendezvous Position the robots s.t. they are
mutually visible to each other.

• Solving rendezvous is easy!
• 1. Compute minimum-base.
• 2. Identify C
• 3. Go to any vertex of C

19
Constructing Gvis

• Edges incident to C
• Can be identified easily.
• Clique edges partitions P
• Each class appears once in each part
• Same holds for any other class that forms a clique

How to identify the remaining edges?
20
Identifying Adjacencies

• Identify edges (vi,vik) of increasing distances
  k 2, 3, ..., n/2.
• Is the next unidentified vertex
• vj vik or not?
• Easy, if in different classes.
• Let y be the number of dist. (k-1) backward-edges
  of vik
• Go to vj and look back (LB)

We can show that vj vik ltgt LB -(y1)
21
Complexity of the Algorithm

• The complexity is dominated by cost of
  constructing minimum-base
• Walk along the boundary and identify the
  neighbors of each vertex.
• Use distinguishing paths to identify classes (n-1
  paths of length n)
• Iteratively obtain distinguishing paths for k 1
  to n
• Cost O(n3m) moves.
• (Additional cost O(n2) moves)

22
Summary

• A robot moving in a polygon P that
• knows an UB on n (vertices of P)
• and can look back
• Is able to
• compute the value of n
• construct the visibility graph
• solve rendezvous
• Why visibility graphs?
• It is not possible to determine the exact shape
  of the polygon.
• Visibility graphs provide sufficient topological
  information.

23
Related Results

• With angle measurements at each vertex
• A robot moving only on the boundary and knowing
  n, can reconstruct the polygon. Disser et al.
  2010
• Convexity Detection Look-back
• A robot knowing n can construct the visibility
  graph Bilo et al. 2009
• Only Convexity Detection
• A robot can knowing n construct the visibility
  graph (in exponential time) (unpublished)
• Impossibility
• A robot moving on the boundary can not construct
  visibility graph even if it knows n Bilo et al.
  2009
• Distance Measurements to visible vertices
• Can the robot construct the visibility graph?

24
Merci de votre attention!