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

2. 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. 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. 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. 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. 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. 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

8. 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. 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. 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. 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 i ff A(P) = 2r + 1

12. 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. 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. 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. 15 1 2 3 4 5 6 123456123456 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 v 2,2 and v 5,5 Q0Q0 Q1Q1

16. 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: v 2+3i, 2+3i, i = 0, 1, …, k

17. 17 Guarding MIN-AREA grid n -ogons 1 2 3 4 5 6 123456123456 123456123456 MIN-AREA grid n -ogon with r = 7 1 2 3 4 5 6 7 8 9 123456789123456789 1 2 3 4 5 6 123456123456

18. 18 Guarding MIN-AREA grid n -ogons P1P1 1 2 3 4 5 6 7 8 9 123456789123456789

19. 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 = 1r = 2r = 3 r = 4r = 5r = 6

20. 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. 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 MA r be the value of the area of a THIN-MAX- AREA grid n -ogon with r reflex vertices

22. 22 Other classes of THIN grid n -ogons By observation, we concluded, that Conjecture: For r ≥ 6, MA 2 = 6MA 3 = 11MA 4 = 17MA 5 = 24 MA 3 = MA 2 + 5 MA 4 = MA 3 + 6 = MA 2 + 5 + 6 MA 5 = MA 4 + 7 = MA 2 + 5 + 6 + 7

23. 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. 24 Ana Mafalda Martins ammartins@crb.ucp.pt Thanks four your attention

25. 25 Introduction  Minimum Vertex Guard (MVG) Problem

26. 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. 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. 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. 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. 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: k = 2 MIN-AREA n-ogon with r = 7 Proof

31. 31 Guarding MIN-AREA grid n-ogons vg: v 2,2, v 5,5, v 8,8

32. 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. 33 Guarding MIN-AREA grid n-ogons r p = r q + 3=3k + 4 A(P) = A(Q) + 6 = 2r q + 1 + 6 = 2(r p -3) + 7 = 2 r p +1 I.H. MIN-AREA r q = 3k + 1 MIN-AREA 12 - ogon

34. 34 Guarding MIN-AREA grid n-ogons H.I. vg = k + 1 v 2,2, v 5,5,..., v 2+3k, 2+3k vg = (k + 1) + 1 = k + 2 v 2,2, v 5,5,..., v 2+3k, 2+3k and v 5+3k, 5+3k

35. 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.