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Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais. Ana Mafalda Martins Universidade Católica Portuguesa CEOC. Encontro Anual CEOC e CIMA-UE. How many guards* are always sufficient to guard any simple polygon P with n vertices?. - PowerPoint PPT Presentation
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Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais
Ana Mafalda MartinsUniversidade Católica
PortuguesaCEOC
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?
How many guards* are always sufficient to
guard any simple polygon P with n vertices?
* Each guard is stationed at a fixed point, has 2 range visibility, and cannot see trough the 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 P’s boundary
we denote: this partition by Π(P) and the number of its elements (pieces) by |Π(P)|
since each piece is a rectangle, we call it a r-piece
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 |Π(Q)| |Π(P)|, for all grid
n-ogons P
Similarly, a grid n-ogon Q is called THIN iff |Π(Q)| |Π(P)|, for all grid n-ogons P
O’Rourke 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 then12|)(| rP
[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
odd for ,
4
)1(3
even for ,4
463
)(2
2
rr
rrr
P
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 = (r+2)/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
1 2 3 4 5 6
1
2
3
4
5
6
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
Q0
Q1
16
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 = (r+2)/3 = n/6 vertex guards are necessary to cover it
and those vertex guards are: v2+3i, 2+3i , i = 0, 1, …,
k
17
Guarding MIN-AREA grid n-ogons
1 2 3 4 5 6123456
1 2 3 4 5 6123456
MIN-AREA grid n-ogonwith r = 7
1 2 3 4 5 6 7 8 9
123456789
1 2 3 4 5 6123456
18
Guarding MIN-AREA grid n-ogons
P1
1 2 3 4 5 6 7 8 91
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
20
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
2
25
)2(7652
2
rr
rMAMAr
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 (…)
…
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
[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
Max-Area n-ogons, for n = 16
29
Conventions, Definitions and Results
FATs are NOT the MAX-AREA grid n-ogon
FAT grid 14-ogon, A(P) = 27
“NOT” FAT grid 14-ogon, A(P) = 28
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: kiv ii ,...,1,0,32,32
k = 2
MIN-AREA n-ogonwith r = 7
Proof
31
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 k+1
First, we must prove that “merging” k+1 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
rp = rq+ 3=3k + 4
A(P) = A(Q) + 6 = 2rq + 1 + 6 = 2(rp-3) + 7 = 2 rp+1
I.H.MIN-AREA rq= 3k + 1
MIN-AREA 12 - ogon
34
Guarding MIN-AREA grid n-ogons
H.I. vg = k + 1v2,2, v5,5,..., v2+3k, 2+3k
vg = (k + 1) + 1 = k + 2v2,2, v5,5,..., v2+3k, 2+3k and v5+3k, 5+3k
35
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.