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Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais

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?.

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Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais

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  1. Caracterização e Vigilância de algumas Subclasses de Polígonos Ortogonais Ana MafaldaMartins Universidade Católica Portuguesa CEOC Encontro Anual CEOC e CIMA-UE

  2. How many guards* are always sufficient to guard any simple polygon Pwith nvertices? * Each guard is stationed at a fixed point, has 2 range visibility, and cannot see trough the walls 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 nwalls?

  3. Introduction • Soon, in 1975, Chvátal proved the well known Chvátal Art Gallery Theorem: • n/3 guards are occasionally necessary and alwayssufficient to cover asimple polygonof nvertices • Avis and Toussaint (1981) developed anO(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 ofminimizing the number of guards necessary to cover a given simple polygon P, arbitrary or orthogonal, is showed to beNP-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 gridn-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 an-ogonPis obtained by extending each edge incident to a reflex vertex of P towards the interior ofPuntil it hits P’sboundary • 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

  8. Conventions, Definitions and Results • Definition: An-ogonPis in general position iffP has no collinear edges • Definition: A grid n-ogonis an-ogonin general position defined in a (n/2)x(n/2) square grid • Definition:Agrid n-ogon Qis calledFATiff|Π(Q)| |Π(P)|, for allgrid n-ogons P Similarly, agrid n-ogon Qis called THINiff|Π(Q)| |Π(P)|, for allgrid n-ogonsP • 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 Pis 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 THINgridn-ogonsare NOT unique THIN 10-ogons

  11. Conventions, Definitions and Results • The area A(P)of agrid n- ogon Pis the number of grid cells in its interior • Proposition: Let Pbe a gridn-ogon withrreflex vertices; then 2r +1A(P)  r 2 + 3 • Definition: A gridn-ogon is a: • MAX-AREA gridn-ogoniff A(P) = r 2 + 3and • MIN-AREA gridn-ogoniff A(P) = 2r +1

  12. Conventions, Definitions and Results • There existMAX-AREA grid n-ogons for all n;however they are not unique • FATsare NOT the MAX-AREAgridn-ogons • There is a single MIN-AREA grid n-ogon (symmetries excluded) • All MIN-AREA are THIN; but, NOT allTHIN 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 gridn-ogons • We think that FATs and THINs can be representative of extreme behaviour • Problem: Given a FAT or a THIN gridn-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-AREAgridn-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. Q1 Guarding MIN-AREA grid n-ogons • Lemma:Twovertex guards are necessary to covertheMIN-AREA12-ogon (r = 4). Moreover, the only way to do so is with the vertex guardsv2,2andv5,5 1 2 3 4 5 6 1 2 3 4 5 6 Q0

  16. Guarding MIN-AREA grid n-ogons • Proposition: LetPbe aMIN-AREA gridn-ogonwithr≥ 7reflex vertices andr =3k + 1 then: • we can obtain it “merging” k = (r-1)/3MIN-AREA 12-ogons • k + 1 = (r+2)/3 = n/6 vertex guards are necessaryto cover it • and those vertex guards are: v2+3i, 2+3i , i = 0, 1, …, k

  17. MIN-AREA grid n-ogon with r = 7 1 2 3 4 5 6 123456 1 2 3 4 5 6 7 8 9 123456 7 8 9 1 2 3 4 5 6 1 2 3 4 5 6 123456 123456 Guarding MIN-AREA grid n-ogons

  18. P1 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 rreflex vertices r = 1 r = 2 r = 3 r = 4 r = 5 r = 6

  20. Other classes of THIN grid n-ogons • Definition: A gridn-ogon is called SPIRALif its boundary can be divided into a reflex chain and a convex chain • Some results: • SPIRAL gridn-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 gridn-ogon with maximum area(THIN-MAX-AREA grid n-ogon)? • Let MArbe the value of the area of a THIN-MAX-AREA grid n-ogon with rreflex vertices

  22. Other classes of THIN grid n-ogons • By observation, we concluded, that • Conjecture: Forr≥ 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 characterizedtwo 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@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-ogonsin general position that are symmetrically equivalent, the number of classes will be further reduced. • In this way, the grid n-ogonin the above figure represent the same class.

  28. Conventions, Definitions and Results • In [1] it is proved that • There exist MAX-AREAgrid n-ogon for all n • However, they are not unique Max-Arean-ogons, forn = 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. FAT grid 14-ogon,A(P) = 27 “NOT” FAT grid 14-ogon,A(P) = 28 Conventions, Definitions and Results • FATsare NOT theMAX-AREA grid n-ogon

  30. Guarding MIN-AREAgrid 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 withr = 7 k= 2

  31. Guarding MIN-AREAgrid n-ogons vg: v2,2 , v5,5 , v8,8

  32. Guarding MIN-AREAgrid n-ogons • Letk≥ 2 • Induction Hypothesis:The proposition is true fork • Induction Thesis: The proposition is true for k+1 • First, we must prove that “merging” k+1MIN-AREAgrid n-ogonwe will obtain the MIN-AREA gridn-ogonwithr =3k +4reflex vertices

  33. Guarding MIN-AREAgrid n-ogons I.H. MIN-AREA rq= 3k + 1 MIN-AREA 12 - ogon rp = rq+ 3=3k + 4 A(P) = A(Q) + 6= 2rq + 1 + 6 = 2(rp-3) + 7 = 2 rp+1

  34. Guarding MIN-AREAgrid n-ogons H.I. vg = k + 1 v2,2, v5,5,..., v2+3k, 2+3k vg= (k + 1) + 1 = k + 2 v2,2, v5,5,..., v2+3k, 2+3kandv5+3k, 5+3k

  35. Guarding Fat & Thin grid n-ogons • We already proved, in [2], that to cover a FAT To guard completely any FATgrid n-ogon it is always sufficient two /2vertex 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.

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