Caracteriza

1
Caracterização e Vigilância de algumas Subclasses
de Polígonos Ortogonais
Ana Mafalda Martins Universidade Católica
Portuguesa CEOC
Encontro Anual CEOC e CIMA-UE
2
Introduction

• Victor Klee, in 1973, posed the following problem
  to Vasek Chvátal

How many guards are enough to cover the interior
of an art gallery room with n walls?
3
Introduction

• Soon, in 1975, Chvátal proved the well known
  Chvátal Art Gallery Theorem
• ?n/3? guards are occasionally necessary and
  always sufficient to cover a simple polygon of n
  vertices
• Avis and Toussaint (1981) developed an O(nlogn)
  time algorithm for locating ?n/3? guards in a
  simple polygon

4
Introduction

• For orthogonal polygons, Kahn et al. (1983) have
  shown that
• ?n/4? guards are occasionally necessary and
  always sufficient to cover an orthogonal polygon
  of n vertices (n-ogon)
• The problem of minimizing the number of guards
  necessary to cover a given simple polygon P,
  arbitrary or orthogonal, is showed to be NP-Hard!

5
Introduction

• Minimum Vertex Guard (MVG) Problem given a
  simple polygon P, find the minimum number of
  guards placed on vertices (vertex guards)
  necessary to cover P

6
Introduction

• Our contribution
• we will introduce a subclass of orthogonal
  polygons the grid n-ogons,
• study and formalize their characteristics, in
  particular, the way they can be guarded with
  vertex guards

7
Conventions, Definitions and Results

• Definition A rectilinear cut (r-cut) of a n-ogon
  P is obtained by extending each edge incident to
  a reflex vertex of P towards the interior of P
  until it hits Ps boundary
• we denote
• this partition by ?(P) and
• the number of its elements (pieces) by ?(P)
• since each piece is a rectangle, we call it a
  r-piece

8
Conventions, Definitions and Results

• Definition A n-ogon P is in general position iff
  P has no collinear edges
• Definition A grid n-ogon is a n-ogon in general
  position defined in a (n/2)x(n/2) square grid
• Definition A grid n-ogon Q is called FAT
  iff ?(Q) ? ?(P), for all grid n-ogons P
• Similarly, a grid n-ogon Q is called THIN iff
  ?(Q) ? ?(P), for all grid n-ogons P
• ORourke proved that n 2r 4, for all n-ogon

9
Conventions, Definitions and Results

• Let P be a grid n-ogon and r (n - 4)/2 the
  number of its reflex vertices. In 1 it is
  proved that
• If P is FAT then
• If P is THIN then

1 Bajuelos A.L, Tomás A. P., Marques F.,
Partitioning Polygons by Extension of All Edges
Incident to Reflex Vertices lower and upper
bound on the number of pieces. ICCSA 2003
10
Conventions, Definitions and Results

• There is a single FAT grid n-ogon (symmetries
  excluded) and its form is illustrated in the
  following figure
• The THIN grid n-ogons are NOT unique

THIN 10-ogons
11
Conventions, Definitions and Results

• The area A(P) of a grid n- ogon P is the number
  of grid cells in its interior
• Proposition Let P be a grid n-ogon with r
  reflex vertices then 2r 1 ? A(P) ? r 2 3
• Definition A grid n-ogon is a
• MAX-AREA grid n-ogon iff A(P) r 2 3 and
• MIN-AREA grid n-ogon iff A(P) 2r 1

12
Conventions, Definitions and Results

• There exist MAX-AREA grid n-ogons for all n
  however they are not unique
• FATs are NOT the MAX-AREA grid n-ogons
• There is a single MIN-AREA grid n-ogon
  (symmetries excluded)
• All MIN-AREA are THIN but, NOT all THIN are
  MIN-AREA

THIN grid 12-ogon, A(P) 15
13
Guarding FAT and THIN grid n-ogons

• Our main goal is to study the MVG problem for
  grid n-ogons
• We think that FATs and THINs can be
  representative of extreme behaviour
• Problem Given a FAT or a THIN grid n-ogon,
• determine the number of vertex guards necessary
  to cover it
• and where these guards must be placed

14
Guarding FAT and THIN grid n-ogons

• For FATs the problem is already solved (2)
• The THINs are not so easier to cover
• Up to now, the only quite characterized subclass
  of THINs is the MIN-AREA grid n-ogon
• We already proved that
• ?n/6? ?(r2)/3? vertex guards are always
  sufficient to cover a MIN-AREA grid n-ogon (2)
• We prove now that this number is in fact
  necessary and we establish a possible positioning

2 Martins, A.M., Bajuelos A.L, Some
properties of FAT and THIN grid n-ogons. ICNAAM
2005.
15
Guarding MIN-AREA grid n-ogons

• Lemma Two vertex guards are necessary to cover
  the MIN-AREA 12-ogon (r 4). Moreover, the only
  way to do so is with the vertex guards v2,2 and
  v5,5

1 2 3 4 5 6
1 2 3 4 5 6
Q0
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Guarding MIN-AREA grid n-ogons

• Proposition Let P be a MIN-AREA grid n-ogon with
  r 7 reflex vertices and r 3k 1 then
• we can obtain it merging k (r-1)/3 MIN-AREA
  12-ogons
• k 1 ?(r2)/3? ?n/6? vertex guards are
  necessary to cover it
• and those vertex guards are v23i, 23i , i
  0, 1, , k

17
Guarding MIN-AREA grid n-ogons

18
Guarding MIN-AREA grid n-ogons
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
19
Guarding MIN-AREA grid n-ogons

• Proposition ?(r 2) / 3? ?n / 6? vertex
  guards are always necessary to cover any MIN-AREA
  grid n-ogon with r reflex vertices

r 1
r 2
r 3
r 4
r 5
r 6
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Other classes of THIN grid n-ogons

• Definition A grid n-ogon is called SPIRAL if its
  boundary can be divided into a reflex chain and a
  convex chain
• Some results
• SPIRAL grid n-ogon is a THIN grid n-ogon
• ?n/4? vertex guards are necessary to cover a
  SPIRAL grid n-ogon

21
Other classes of THIN grid n-ogons

• What is the value of the area of a THIN grid
  n-ogon with maximum area (THIN-MAX-AREA grid
  n-ogon)?
• Let MAr be the value of the area of a
  THIN-MAX-AREA grid n-ogon with r reflex vertices

22
Other classes of THIN grid n-ogons

• By observation, we concluded, that
• Conjecture For r 6,

MA2 6
MA3 11
MA4 17
MA5 24
MA3 MA2 5
MA4 MA3 6 MA2 5 6
MA5 MA4 7 MA2 5 6 7
23
Conclusions and Further Work

• We defined a particular type of orthogonal
  polygons the grid n-ogons
• With the aim of solving the MVG problem for
  THINs,
• we already characterized two classes of THINs
• MIN-AREA grid n-ogons
• SPIRAL grid n-ogons
• we are characterizing
• THIN-MAX-AREA grid n-ogons ()

24

Thanks four your attention
Ana Mafalda Martins ammartins_at_crb.ucp.pt
25
Introduction

• Minimum Vertex Guard (MVG) Problem

26
Conventions, Definitions e Results

• Each n-ogon in general position is mapped to a
  unique grid n-ogon trough top-to-bottom and
  left-to-right sweep.
• Reciprocally, given a grid n-ogon we may
  create a n-ogon that is an instance of its class
  by randomly spacing the grid lines in such a way
  that their relative order is kept.

27
Conventions, Definitions and Results

• If we group grid n-ogons in general position that
  are symmetrically equivalent, the number of
  classes will be further reduced.
• In this way, the grid n-ogon in the above figure
  represent the same class.

28
Conventions, Definitions and Results

• In 1 it is proved that
• There exist MAX-AREA grid n-ogon for all n
• However, they are not unique

Max-Area n-ogons, for n 16
1 Bajuelos A.L, Tomás A. P., Marques F.,
Partitioning Polygons by Extension of All Edges
Incident to Reflex Vertices lower and upper
bound on the number of pieces. ICCSA 2003
29
Conventions, Definitions and Results

• FATs are NOT the MAX-AREA grid n-ogon

30
Guarding MIN-AREA grid n-ogons

• Proposition Merging k 2 MIN-AREA 12-ogons
  we will obtain the MIN-AREA grid n-ogon with r
  3k 1. More, k 1 vertex guards are necessary
  to cover it, and the only way to do so is with
  the vertex guards

Proof
MIN-AREA n-ogon with r 7
k 2
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Guarding MIN-AREA grid n-ogons
vg v2,2 , v5,5 , v8,8
32
Guarding MIN-AREA grid n-ogons

• Let k 2
• Induction Hypothesis The proposition is true for
  k
• Induction Thesis The proposition is true for k1
• First, we must prove that merging k1 MIN-AREA
  grid n-ogon we will obtain the MIN-AREA grid
  n-ogon with r 3k 4 reflex vertices

33
Guarding MIN-AREA grid n-ogons
I.H. MIN-AREA rq 3k 1
MIN-AREA 12 - ogon
rp rq 33k 4
A(P) A(Q) 6 2rq 1 6
2(rp-3) 7 2 rp1
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Guarding MIN-AREA grid n-ogons
H.I.
vg k 1 v2,2, v5,5,..., v23k, 23k
vg (k 1) 1 k 2 v2,2, v5,5,..., v23k,
23k and v53k, 53k
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Guarding Fat Thin grid n-ogons

• We already proved, in 2, that to cover a FAT To
  guard completely any FAT grid n-ogon it is always
  sufficient two ?/2 vertex guards, and
  established where they must be placed

Vertex guards with ?/2 range visibility
2 Martins, A.M., Bajuelos A.L, Some
properties of FAT and THIN grid n-ogons. ICNAAM
2005.