Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais Ana Mafalda Martins Universidade Católica Portuguesa CEOC Encontro Anual CEOC.

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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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


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


P1

1 2 3 4 5 6 7 8 91

2

3

4

5

6

7

8

9

19


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

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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

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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

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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 (…)

…

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Ana Mafalda Martins

[email protected]

Thanks four your attention

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


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.

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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

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FATs are NOT the MAX-AREA grid n-ogon

FAT grid 14-ogon, A(P) = 27

“NOT” FAT grid 14-ogon, A(P) = 28

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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

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vg: v2,2 , v5,5 , v8,8

32


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

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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

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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

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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.

Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais Ana Mafalda Martins Universidade Católica Portuguesa CEOC Encontro Anual CEOC.

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