Coloring Variations of the Art Gallery Problem

Transcription

1 Coloring Variations of the Art Gallery Problem Master s Thesis Andreas Bärtschi August 2011 Advisors: Prof. Dr. S. Suri (UCSB), Prof. Dr. E. Welzl (ETHZ) Department of Mathematics, ETH Zürich

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3 Abstract The art gallery problem [2] asks for the smallest possible size of a point set S (the guards) to completely guard the interior of a simple polygon P (the art gallery). This thesis treats variations of the original problem that arise when we introduce a coloring of the guards. Rather than asking for the minimum number of guards we ask for the minimum number of colors. In [4] L. H. Erickson and S. M. LaValle introduced the following coloring: Two guards should be given different colors if their visibility regions intersect. What is the minimum number of colors required to color any guard set of P? We call this number the strong chromatic guard number of P, χ sg (P). In the first part of the thesis (chapter 3) we show that for all positive integers n: - there exists a polygon P n with n vertices and χ sg (P n ) = Ω(n) and - an orthogonal polygon P n with n vertices and χ sg (P n ) = Ω( n). In the main part of the thesis (chapters 4 and 5) we show that when slightly changing the problem definition by using a conflict-free coloring of the guards, we can get polylogarithmic upper bounds for the conflictfree chromatic guard number χ cfg (P): - for all orthogonal polygon P n with n vertices, χ cfg (P n ) = O(log n) - and for all polygon P n with n vertices, χ cfg (P n ) = O((log n) 2 ). i

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5 Contents Contents iii 1 Introduction 1 2 Chromatic art gallery problems Visibility in polygons A first chromatic art gallery problem Visibility hypergraphs Chromatic art gallery problems Strong guard coloring in polygons Motivation Upper bounds A constant upper bound for staircase polygons A constant upper bound for spiral polygons Lower bounds A lower bound for monotone orthogonal polygons A lower bound for general polygons Conflict-free guard coloring in orthogonal polygons Motivation The building blocks Dividing into four equivalent subproblems A necessary condition for intersections Conquering red-black-alternating subtrees A logarithmic upper bound for orthogonal polygons Conflict-free guard coloring in general polygons A logarithmic upper bound for monotone polygons Polygon partitioning revisited A polylogarithmic upper bound for general polygons iii

6 Contents 6 Conclusion 55 Bibliography 57 iv

7 Chapter 1 Introduction Since Steve Fisk s elegant proof [8] that n 3 guards always suffice to guard the interior of a simple polygon, many variations of the original art gallery problem (also known as Chvátal s watchman theorem) have been studied. Results include a better bound for orthogonal polygons, bounds for covering the exterior of a polygon, and progress on visibility graphs. An entire book by Joseph O Rourke can be found in [10]; there is also a good overview by Thomas Shermer [12]. Somewhat surprisingly Fisk s proof uses a coloring of the vertices of a polygon until recently no work has been done that asks for a coloring of the guards. Coloring of the guards In a guard set S of a polygon P each guard p covers the area it sees inside P, i.e. its visibility region V(p). These regions intersect, raising a natural connection to hypergraphs: Let H(S, P) denote the hypergraph that consists of vertices that correspond to the guards in S and edges that correspond to the intersections of the visibility regions of these guards. p 1 p 2 p 3 H(S, P ) p 2 P p 1 p 3 Figure 1.1: A polygon with its guard set and the corresponding hypergraph. In the example in Figure 1.1 H(S, P) contains the vertices p 1, p 2, p 3 and the edges {p 1, p 2 }, {p 2, p 3 }, {p 1, p 2, p 3 } but not the edge {p 1, p 3 } since V(p 1 ) V(p 3 ) is a subset of V(p 2 ). 1

8 1. Introduction This allows us to translate colorings of hypergraphs to colorings of guards. In this thesis we look at strong coloring and conflict-free coloring. Lawrence Erickson and Steven LaValle first suggested in [4] to ask for the strong chromatic guard number of a polygon P, denoted by χ sg (P) 1 : Rather than determining the strong chromatic number of H(S, P) for a given guard set S, we look for the minimum number of colors to color any guard set such that the resulting hypergraph has a strong coloring with χ sg (P) colors. Clearly, for a polygon P n on n vertices, we have χ sg (P n ) n 3 by Chvátal s watchman theorem (we can give each guard a unique color). Apart from this relation, the chromatic problem differs a lot from the original art gallery problem, since we are not interested in the minimum number of guards but the minimum number of colors. The number of necessary colors can deviate by a large factor from the number of guards necessary, as we show in Figure 1.2. Nonetheless we discovered that for all n there exist Figure 1.2: The standard comb polygon requires n 3 guards but only two colors for a strong guard coloring of the given guard set. polygons P n, for which χ sg (P n ) is up to a constant factor not better than the trivial solution n 3 (see Chapter 3). Erickson and LaValle independently gave a similar result in [3]. Contribution of this thesis In this thesis we show that a conflict-free coloring (CF-coloring for short) rather than a strong coloring of the guards gives us very good upper bounds. In a CF-coloring of the guards every edge in the hypergraph H(S, P) has a vertex of unique color among its vertices. This means that each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. This is not only interesting from a theoretical point of view but CF-coloring has many possible applications in wireless communication and related fields. Upper and lower bounds for CF-coloring in hypergraphs have been studied for hypergraphs resulting from geometric objects such as discrete intervals, pseudo-disks [14] and axisparallel rectangles [11, 14]. Algorithmic approaches include approximation and online algorithms. 1 Erickson and LaValle used the term chromatic guard number whereas we use the term strong chromatic guard number to distinguish between different colorings. 2

9 To the best of our knowledge there aren t any results for CF-coloring of visibility regions in polygons so far. The main part of this thesis (chapters 4 and 5) establishes polylogarithmic upper bounds for the conflict-free chromatic guard number χ cfg. For polygons on n vertices we establish χ cfg (P n ) = O(log n) χ cfg (P n ) = O((log n) 2 ) for orthogonal polygons and for general polygons. Acknowledgments I would like to thank my advisor Prof. Subhash Suri for his great support, his suggestions to my work and for hosting me at UCSB. I also enjoyed the many interesting and helpful discussions with Luca Foschini. I am very grateful to Prof. Emo Welzl for supervising my thesis on behalf of ETH and to ETH Zürich for enabling this thesis by admitting me to the exchange program. 3

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11 Chapter 2 Chromatic art gallery problems In this chapter we introduce notations and concepts that we are going to use in the subsequent chapters. Furthermore we define two chromatic art gallery problems. 2.1 Visibility in polygons Let a polygon P be a closed, simply connected subset of R 2 with boundary P, a closed polygonal chain. Throughout this thesis the polygons we consider are simple, meaning only consecutive line segments (or edges) of P intersect and those only at their endpoints. Sometimes we look at the special case of orthogonal polygons, i.e. polygons whose edges meet at right angles. Let P n denote a polygon with n vertices. Visibility A point q P is visible from a point p P if the closed line segment pq is a subset of P. The visibility region of p, denoted by V(p), is the set of all points visible from p, V(p) := {q P q is visible from p }. Guard sets A finite point set S is called a guard set of P if each point in P is seen by at least one of the elements of S, i.e. if p S V(p) = P. We refer to a member of S as guard. By abuse of notation we call a pair of guards p, q S intersecting if their visibility regions V(p) and V(q) intersect. 2-link-visibility We are interested in how guards in S intersect. One way to explore this is by looking at the link distance LD(p, q) of two guards p and q. LD(p, q) is the minimum length of any polygonal path connecting p and q in P (each line segment of the path must be contained in P). If LD(p, q) 2, we call p and q 2-link-visible. Proposition 2.1 Two guards p and q are 2-link-visible if and only if they intersect. 5

12 2. Chromatic art gallery problems Proof This follows immediately from the definitions. 2-link-visibility graph In [5] Erickson and LaValle have introduced 2-linkvisibility graphs. The idea of a 2-link-visibility graph of a polygon P and a guard set S is the following: the vertices should correspond to the guards in S, and two vertices should be connected if and only if the corresponding guards are 2-link-visible in P. Definition 2.2 Let P be a polygon and S a guard set. Then the 2-link-visibility graph of S in P, denoted by G(S, P), is defined as G(S, P) := (S, {{p, q} p, q S and LD(p, q) 2}). In other words, a 2-link-visibility graph G(S, P) is the intersection graph of the visibility regions of the guards in S (by Proposition 2.1). Suppose we want to find the chromatic number χ of G(S, P) for a given guard set S. This would be easy if 2-link-visibility graphs formed a subclass of a class of graphs with known polynomial-time algorithms to find χ. In [9] for example it was shown that one can obtain a polynomial algorithm to find a minimum coloring of perfect graphs. Chordal graphs are a subclass of both perfect graphs and polygon-circle graphs. Since the guards visibility regions are polygons with vertices on the same simple polygonal chain, 2-link-visibilitygraphs are a subclass of polygon-circle graphs as well. However, we have found counterexamples to 2-link-visibility-graphs being chordal, and they are also not necessarily perfect. Apart from that, no properties of the class of 2-link-visibility graphs are known to the best of my knowledge, especially no polynomial algorithms to find the chromatic number. 2.2 A first chromatic art gallery problem In 2010 Erickson and LaValle posed the chromatic art gallery problem in [4], which introduced for the first time a question on coloring guards in P. Their problem definition was the following: Let S be a guard set of P. Two guards should be given different colors if they intersect. Let C(S) be the minimum number of colors required to color S in this manner. Let T(P) be the set of all guard sets of P. Then we can define the chromatic guard number of P, denoted by χ G (P): χ G (P) := min S T(P) C(S). Possible questions include asking for the chromatic guard number for specific shapes of polygons, for example staircase and spiral polygons, or giving bounds for χ G (P n ) depending on the number of vertices n of a polygon. A coloring of a guard set is based on a vertex coloring of the underlying 2- link-visibility graph, namely we have C(S) = χ(g(s, P)). In order to acquire 6

13 2.3. Visibility hypergraphs a better understanding of how guards can intersect and to explore other possible coloring notions, it is interesting to look how much information about guard intersections we can get from a 2-link-visibility graph. Does a 2-linkvisibility graph represent the complete structure of the intersections of visibility regions? For example, three guards may have a common intersection and thus induce a three-vertex clique in G(S, P). Is there for every pair of them also an intersection of the visibility regions that is not covered by the third guard? This is not necessarily the case, as we can see in Figure 2.1: Both guard sets give rise to the same 2-link-visibility graph but in the guard set to the left we have (V(p 1 ) V(p 3 )) \V(p 2 ) = which is not the case for the guard set on the right. p 1 p 2 p 3 p 2 p 1 p 2 p 3 P p 1 G(S, P ) p 3 P Figure 2.1: Two guard sets that have different intersections of their guards visibility regions but the same induced 2-link-visibility graph. In the following section we want to look at another way to represent the structure of visibility regions and their intersections: hypergraphs. 2.3 Visibility hypergraphs Let S be a guard set of P. For a point q P we define l(q) to be the list of all guards that contain q in their visibility region, l(q) := {p S q V(p) }. The idea of a visibility hypergraph of S in P is the following: the vertices should correspond to the guards in S, and a subset of the vertices defines an edge if and only if the corresponding guards visibility regions have a common intersection that is not seen by any other guard. Definition 2.3 Let P be a polygon and S a guard set. Then the visibility hypergraph of S in P, denoted by H(S, P), is defined as H(S, P) := (S, {l(q) q P }). For hypergraphs we have many different notions of vertex coloring, for example proper coloring, strong coloring, conflict-free coloring and uniquemaximum coloring. Basically all of these colorings can be translated to a chromatic art gallery problem; in this thesis we focus on strong coloring and conflict-free coloring. For the following definitions, let H = (V, E) be a hypergraph with a vertex set V and an edge set E 2 V. 7

14 2. Chromatic art gallery problems Definition 2.4 (strong coloring) A strong coloring of a hypergraph H is an assignment of colors to the vertices of H such that in every edge e E all vertices of e have different colors. The strong chromatic number χ s (H) is the minimum number of colors for which H admits a strong coloring. Lemma 2.5 Let S be a guard set of a polygon P. Then a strong coloring of the visibility hypergraph H(S, P) can be viewed as a regular vertex coloring of the 2- link-visibility graph G(S, P) 1. In particular, χ s (H(S, P)) = χ(g(s, P)). Proof Clearly both graphs have the same vertex set, V(H) = S = V(G). Denote the edge set of H by E and the edge set of G by E. Consider two guards p, q S. Assume p and q are adjacent in H, i.e. there exists an edge e E such that p, q e. Therefore V(p) V(q) = by Definition 2.3. But then p and q are adjacent in G and any strong coloring of H is also a vertex coloring of G. Assume on the other hand that p and q are adjacent in G, i.e. there exists an edge e E such that e = {p, q}. Then V(p) V(q) = by Definition 2.2. But then {p, q} e for some e E and p and q are adjacent in H. Thus every vertex coloring of G is also a strong coloring of H. Definition 2.6 (conflict-free coloring) A conflict-free coloring of a hypergraph H is an assignment of colors to the vertices of H such that in every edge e E there exists a vertex with unique color among all the vertices of e. We will also use the term CF-coloring for short. The conflict-free chromatic number χ cf (H) is the minimum number of colors for which H admits a CF-coloring. CF-coloring in hypergraphs has been introduced and studied for the first time by Even et al. [6] and Shakhar Smorodinsky [13]. There has been much progress in this area since its introduction; we want to mention hypergraphs induced by regions particularly. For a summary that covers a lot of the research done see a survey by Smorodinsky [15]. 2.4 Chromatic art gallery problems For any specific type of vertex coloring in hypergraphs we can now define the corresponding coloring of a guard set S in P. We will do this for strong coloring and CF-coloring. In this way we can define a strong chromatic guard number and a conflict-free chromatic guard number, for which we can ask similar questions as for the chromatic guard number. A chromatic art gallery problem is therefore always defined with respect to a specific coloring. 1 G is therefore called the representing graph, 2-section graph or clique graph of H, see [1] 8

15 2.4. Chromatic art gallery problems Strong coloring of the guards A coloring of the guards S in P is a strong coloring of the guards if the same coloring of S in the visibility hypergraph H(S, P) is a strong coloring. Definition 2.7 (strong chromatic guard number) Let C s (S) be the minimum number of colors required to color S with a strong coloring. Let T(P) be the set of all guard sets of P. Then we can define the strong chromatic guard number of P, denoted by χ sg (P): χ sg (P) := min S T(P) C s(s). By definition we have C s (S) = χ s (H(S, P)) and thus χ sg (P) = min S T(P) χ s(h(s, P)). By Lemma 2.5 on the facing page the chromatic art gallery problem with a strong coloring of the guards is therefore equivalent to the one given by Erickson and LaValle in their original paper. Nonetheless we will use the notation of strong coloring throughout this thesis to distinguish from the chromatic art gallery problem with a conflict-free coloring of the guards. CF-coloring of the guards A coloring of the guards S in P is a conflict-free coloring of the guards if the same coloring of S in the visibility hypergraph H(S, P) is a CF-coloring. Definition 2.8 (conflict-free chromatic guard number) Let C cf (S) be the minimum number of colors required to color S with a CF-coloring. Let T(P) be the set of all guard sets of P. Then we can define the conflict-free chromatic guard number of P, denoted by χ cfg (P): χ cfg (P) := min S T(P) C cf(s). Since any strong coloring is also a conflict-free coloring, we get χ cfg (P) χ sg (P). The results in this thesis show that there are polygons on n vertices for which we have an upper bound on χ cfg that is exponentially better than χ sg : In Chapter 3 we give lower bounds for the strong-chromatic guard number χ sg for polygons P n, in particular Ω( n) for orthogonal polygons and Ω(n) for general polygons. In Chapter 4 we give an upper bound of O(log n) for the conflict-free chromatic guard number χ cfg of orthogonal polygons P n. In Chapter 5 we adapt the methods from Chapter 4 to come up with an upper bound of O((log n) 2 ) for the conflict-free chromatic guard number χ cfg of general polygons P n. 9

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17 Chapter 3 Strong guard coloring in polygons In this chapter we present a part of the original paper on a chromatic art gallery problem [4] of Erickson and LaValle, including two upper bounds for the strong chromatic guard number in staircase and spiral polygons. Furthermore we give lower bounds for χ sg for monotone, orthogonal and general polygons. 3.1 Motivation The chromatic art gallery problem with a strong coloring of the guards asks for a placement of colored guards such that each point of the polygon is seen only by guards of pairwise different color. The colors are therefore classes of partially distinguishable guards. Such a problem arises for example from placing landmarks in an environment such that a basic robot can orientate itself: If the robot saw two landmarks of the same color, it may get confused. There are many reasons why one might want to optimize not the number of guards, but rather the number of colors. Assume that the landmarks are implemented as radio beacons: Then we are interested in using only as few frequencies as possible because the usable bandwidth is limited and/or expensive. A strong coloring of the guards may also have further applications in surveillance, sensor networks, and other areas. We are interested in the minimum number of colors needed for such a placement. Does the strong chromatic guard number χ sg grow as we have larger polygons? As it turns out there are constant upper bounds for two special shapes of polygons whereas general and even monotone orthogonal polygons can require a large number of colors. 11

18 3. Strong guard coloring in polygons 3.2 Upper bounds We introduce staircase polygons and spiral polygons and give the guard placement for such polygons found by Erickson and LaValle. The placements allow the use of only three colors (staircase polygons) respectively two colors (spiral polygons) for a strong coloring. The proofs are omitted, an interested reader can find those in [4]. To define the polygon shapes we need the notion of subchains. A subchain is a sequence of consecutive vertices on the boundary of a polygon along with the line segments connecting them. The subchains we are going to look at have a special structure when it comes to the type of vertices they contain. A vertex is called a reflex vertex if the interior angle between its adjacent edges is greater than 180. A convex vertex on the other hand has an interior angle of less than A constant upper bound for staircase polygons A staircase polygon is an orthogonal polygon whose boundary can be separated at two convex vertices v s and v t into two alternating subchains that start with v s and end with v t. An alternating subchain has at least one internal vertex, its first and last internal vertex is convex and the interior vertices are alternatingly convex and reflex vertices. In Figure 3.1 we see a staircase polygon and a placement of guards that needs only three colors, no matter how many vertices the polygon possesses. v s s 1 s 2 s 3 s 4 s 5 s 6 s 1 s 2 s 3 s 4 s 5 s 6 v t s 7 s 7 Figure 3.1: A staircase polygon with a guard placement requiring three colors. Guard placement Assume without loss of generality that the edges are axisparallel and the staircase is descending to the right. Then we can build the guard set S: We place the first guard s 1 on the first convex vertex of the lower subchain from the top. The second guard s 2 is then placed on the right-most vertex on the upper subchain visible from s 1. The position of guard number three, s 3, is then defined as the lowest vertex on the lower subchain visible from s 2. We continue in that manner until we place a guard on the last internal vertex of one of the subchains. 12

19 3.3. Lower bounds Coloring the guards With this guard placement, the polygon is completely covered and no guard s i has an intersection with a guard s j where j > i + 2. In terms of the hypergraph H(S, P) this means that there are only edges of the form {s i, s i+1 }, {s i, s i+2 }, {s i+1, s i+2 } or {s i, s i+1, s i+2 }. Therefore we can color the guards by alternating between three colors and we get Lemma 3.1: Lemma 3.1 For any staircase polygon P n, χ sg (P n ) A constant upper bound for spiral polygons A spiral polygon is a polygon whose boundary can be separated at two convex vertices v s and v t into a convex subchain and a reflex subchain that both start with v s and end with v t. A convex subchain has only convex interior angles while a reflex subchain only has reflex interior vertices, see Figure 3.2 s 2 b 2 s 3 s 2 s 3 b 1 v s s 1 v t s 4 b 3 s 1 Figure 3.2: A spiral polygon with a guard placement requiring two colors. s 4 Guard placement Place the first guard s 1 at the first vertex v s of the convex subchain. Until we have covered the whole polygon we do the following: Let b i be the last vertex on the reflex chain visible from s i. Draw a ray from b i through the next vertex of the reflex chain. Then we place s i+1 at the intersection of the ray with the convex subchain. Coloring the guards With this guard placement, the polygon is completely covered and only consecutive guards intersect. In terms of the hypergraph H(S, P) this means that there are only edges of the form {s i, s i+1 }. Therefore we can color the guards by alternating between two colors and we get Lemma 3.1: Lemma 3.2 For any spiral polygon P n, χ sg (P n ) Lower bounds While the given upper bounds for staircase and spiral polygons looked very promising, it was already shown in [4] that there exist monotone polygons 13

20 3. Strong guard coloring in polygons P n with strong chromatic guard number χ sg (P n ) = Ω( n). First we give a modified version of that construction and its proof to extend this result to monotone orthogonal polygons P n. Then we show an even higher bound for general polygons A lower bound for monotone orthogonal polygons Lemma 3.3 For every positive integer k, there exists an orthogonal polygon P n with n = 4k 2 vertices such that χ sg (P n ) k 4. Proof The polygon P n that we are going to construct is an orthogonal comb that consists of a rectangle of size (2k 2 1) (2k 4) and k 2 evenly spaced notches of size 1 1 on top. The total number of vertices is 4k 2, denoted by n. In Figure 3.3 we show the polygon for k = 6. 1 LHN RHN k = 6 p 2k 4 1 corridors 2k 2 1 Figure 3.3: A monotone orthogonal polygon with 4k 2 vertices for which every guard set needs at least k 4 colors. We split the polygon for our argument into three different regions. The (2k 2 1) (2k 4) rectangle is called the body region of the comb. We divide the notches into the set LHN of the left halves of the notches and the set RHN, the right halves of the notches. We refer to guards in the body region as body guards and to guards in LHN and RHN as apex guards. Visibility of vertices Have a look at the top vertices of the polygon: each notch has two of them, hence there are 2k 2 top vertices. Each apex guard sees exactly the two top vertices in its notch. A body guard can see at most k 1 of the top vertices to its left and k 1 of the top vertices to its right, since a top vertex can only be seen from a point inside a 45 angle (see Figure 3.3). Intersections The right half of each notch extends in a natural way to a corridor of size (1/2) (2k 3). Let s look at a guard p placed in the right half of a notch. The slope of the left boundary of p s visibility region inside the body region is at most 2. Hence no matter where in the notch exactly p 14

21 3.3. Lower bounds is placed, it can see all the points to its left on the lower boundary of P n that are horizontally no further away than k 2 (assuming p is in a notch placed sufficiently far to the right). We make the important observation that p can see at least k 2 1 of the corridors to its left. Hence we can say the following: all apex guards of RHN in consecutive k 2 notches intersect with each other. The same holds for apex guards in LHN. Furthermore all body guards of P n intersect with each other and with all apex guards. Inequalities Let S be a guard set of P n that needs only χ sg (P n ) colors. Let m body be the number of body guards in S and m apex the maximum number of apex guards in either k 2 consecutive notch halves of RHN or k 2 consecutive notch halves of LHN. From the previous remark about intersections we conclude χ sg (P n ) m apex + m body. (3.1) We divide the notches into k2 k/2 = 2k parts of k 2 consecutive notches. Hence we know from the pigeonhole principle that there can be at most 2k m apex guards in RHN and 2k m apex guards in LHN, hence a total of 4k m apex apex guards. On the other hand we have m body body guards. From the remark about the visibility of vertices we know that each apex guard sees 2 top vertices and each body guard at most 2k 2 top vertices. Since S is a guard set, all of the 2k 2 top vertices are indeed covered and we get 2 4k m apex + (2k 2) m body 2k 2. (3.2) Combining the inequalities (3.1) and (3.2) we get χ sg (P n ) m apex + m body m apex + (2k 2) 8k m body k 4, which is the desired lower bound on the number of colors necessary for every guard set. From Lemma 3.3 we have the following corollary: Corollary 3.4 For all positive integers n there exist monotone orthogonal polygons P n with χ sg (P n ) = Ω( n) A lower bound for general polygons For general polygons we found an even higher lower bound. In fact there is only a small constant-factor gap to the trivial coloring solution that arises from a minimum guard set of size n 3 where every guard gets a different color, as we show in Lemma 3.5 on the next page. 15

22 3. Strong guard coloring in polygons Lemma 3.5 For every positive integer k, there exists a polygon P n with n = 3k vertices such that χ sg (P n ) k 2. Proof The idea for constructing such a polygon is the following: On the one hand all guards, no matter where they are placed, should intersect, on the other hand a single guard should only be able to guard a small number of vertices. We construct the polygon P n out of k beams. Take any simple line arrangement of k lines. There are ( k 2 ) intersections of these lines. We can draw a k vertex convex polygon that has all of the line intersections in its interior and exactly one vertex on every line (see Figure 3.4). Now we replace all of these vertices with a long and thin triangle that has one vertex on the corresponding line and the other two vertices very close to the original vertex of the convex polygon. Since every vertex of the convex polygon got replaced with a triangle, we get a polygon P n with n = 3k vertices. k = 4 Figure 3.4: Construction of a polygon with 3k vertices for which every guard set needs at least k 2 colors. 16 We call the new vertices on the lines the notch vertices. There are exactly k of those. Each notch vertex can only be seen by guards inside the extended triangle, colored red in Figure 3.4. We call such a region a beam. For sufficiently long and thin triangles, all the beams intersect pairwise but without an intersection of three beams, since we started the construction with a simple line arrangement. We split the polygon for our argumentation into two different regions. The beams and the rest of the polygon, called the body region. We refer to guards in the body region as body guards and to guards on a beam as beam guards. Visibility of vertices A beam guard can see two notch vertices if and only if it is placed on the intersection of two beams. Otherwise it can see exactly one notch vertex. A body guard only sees the non-notch vertices. Intersections Since all of the beams intersect pairwise, also all the beam guards have to intersect pairwise. Furthermore all the beam guards intersect with all of the body guards.

23 3.3. Lower bounds Inequalities Let S be a guard set of P n that needs only χ sg (P n ) colors. Let m beam be the number of beam guards in S and m body the number of body guards. From the previous remark about intersections we conclude χ sg (P n ) m beam + m body. (3.3) From the remark about the visibility of the notch vertices we know that each beam guard sees at most 2 notch vertices and a body guard doesn t see any notch vertex. Since S is a guard set, all of the k notch vertices are indeed covered and we get Combining the inequalities (3.3) and (3.4) we get 2 m beam k. (3.4) χ sg (P n ) m beam + m body m beam k 2, which is the desired lower bound on the number of colors necessary for every guard set. From Lemma 3.5 we have the following corollary: Corollary 3.6 For all positive integers n there exist polygons P n with a strong chromatic guard number of χ sg (P n ) = Ω(n). For both Lemma 3.3 and Lemma 3.5 the constants can be improved by a slightly more sophisticated construction (and in case of monotone orthogonal polygons a longer proof). Erickson and LaValle have found the bounds of the Corollaries 3.4 and 3.6 independently from my research, their results and proofs with better constants than the ones given here have been published in [5]. 17

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25 Chapter 4 Conflict-free guard coloring in orthogonal polygons In this chapter we give a placement and a conflict-free coloring of guards in orthogonal polygons. We show that O(log n) colors are sufficient in such a placement of colored guards for every orthogonal polygon P n. Throughout this chapter we consider only orthogonal polygons, even if not explicitly mentioned. 4.1 Motivation In the proof of Lemma 3.5 on page 16 we have shown that the strong chromatic guard number can be pretty big. When we focus on a weaker notion of coloring namely conflict-free coloring it turns out that we can get exponentially better bounds compared to strong coloring. Conflict-free coloring is motivated by applications in distributed robotics and wireless sensor networks. Consider for example a setting where the guards correspond to radio towers that are connected by a backbone network. The colors indicate an assignment of frequencies to these towers such that a mobile robot can always communicate with at least one radio tower in its visibility region without interference. To achieve this, there must be at least one guard with a unique color among all guards intersecting at the robots location. Hence contrary to a strong coloring, guards that intersect are allowed to have the same color as long as the intersection of their visibility regions is covered by other guards of unique color. We will show that we can use this property to get a logarithmic upper bound for orthogonal polygons P n on n vertices. 19

26 4. Conflict-free guard coloring in orthogonal polygons 4.2 The building blocks Divide and conquer We will use a divide and conquer approach: We partition orthogonal polygons into four sets of subpolygons; then we show by further dividing each set that we can guard each of the four sets with O(log n) colors. In Figure 4.1 we show an orthogonal polygon and such a partition where the four sets are displayed with four colors. But before we explain our divide and conquer strategy, we introduce the concept of independence and examine two types of polygons that will serve as building blocks in our argument. Figure 4.1: An orthogonal polygon and a partition into four sets of subpolygons. 20 Definition 4.1 (Independence) Let P be a polygon. We call two subpolygons P 1 and P 2 of P independent if there are no points p 1 P 1 and p 2 P 2 that are mutually visible. The key property of two independent subpolygons P 1, P 2 is the following: Let S 1 P 1 and S 2 P 2 be guard sets of P 1 and P 2. Then every pair of guards (s 1, s 2 ) S 1 S 2 intersects only outside of P 1 and P 2. (Assume for the contrary that there exist s 1 S 1, s 2 S 2 such that (V(s 1 ) V(s 2 )) (P 1 P 2 ) =. It follows that V(s 1 ) P 2 = or V(s 2 ) P 1 =, but this contradicts the independence of P 1 and P 2.) Hence we have some freedom in placing guards: Let P 1 be a subpolygon of P with a guard set S 1 P 1 and a conflict-free coloring of S 1 with a color set C. Let P 2 be another subpolygon that is independent from P 1. Then we can place guards in P 2 and color them with colors from C while resting assured that we don t introduce conflicts to P 1 because of the aforementioned property. Definition 4.2 (monotone polygon) A polygon P is called monotone with respect to a line l if every line orthogonal to l intersects the boundary of P at most twice. P is called x-monotone (y-monotone) if P is monotone with respect to the x-axis (respectively the y-axis). According to Definition 4.2, the boundary of an x monotone polygon can be split into an upper subchain and a lower subchain. In the remaining part of Section 4.2 we will look at different kinds of x monotone orthogonal polygons.

27 4.2. The building blocks Staircase polygons One type of our two polygon building blocks are staircase polygons. Staircase polygons are orthogonal polygons that are both x-monotone and y- monotone. In the remarks to Lemma 3.1 on page 13 we gave a placement of guards in staircase polygons and a strong coloring of those guards that needed only three colors. Since any strong coloring is also a conflict-free coloring, we will use the same placement and coloring to achieve a conflict-free coloring of staircase polygons with three colors. Polygons over a single horizontal edge Idea The other kind of polygons that will serve as a building block are monotone orthogonal polygons with the lower subchain of the boundary being a single horizontal edge. We are going to show that such a polygon on n vertices can always be guarded with O(log n) colors. To prove this we first consider the case of two adjacent convex fans, where the upper subchain of the polygon s boundary is the connection of a chain with monotonically increasing y-coordinates of the vertices and a chain with monotonically decreasing y-coordinates of its vertices, see Figure 4.2. Figure 4.2: The union of two convex fans is star-shaped and can be guarded by one guard. In such a polygon P there is a point p visible from all other points q P. Such a polygon is called star-shaped and we can guard it with one guard and one color. In a polygon P we call a star-shaped subpolygon over a single edge with that single edge removed a star pocket. The single edge is removed to expand the notion of independence: We want to consider star pockets that are independent, while the single horizontal edges cutting them off from P don t necessarily need to be independent. If we have k independent star pockets, we can guard these regions with k guards and still use only one color. In Figure 4.3 on the following page we show a partition of an x monotone orthogonal polygon over a single horizontal edge. The polygon is partitioned into three sets of independent star pockets and in each star pocket or on its cut-edge there is a guard. The color of a star pocket indicates the colored guard that has a unique color for any point in that region. 21

28 4. Conflict-free guard coloring in orthogonal polygons Figure 4.3: Independent star pockets of a polygon P and a partition into three sets of independent star pockets. Algorithm We present an algorithm that guards any monotone orthogonal polygon over a single edge with at most log n colors. Given such a polygon, our algorithm should do the following: Search for as many independent star pockets as possible, guard and color them and repeat this for the rest of the polygon. We have already seen how to cover a star pocket, Algorithm 4.1 does exactly this. Algorithm 4.1 GuardStar(Polygon P, Color k) Require: A star-shaped polygon P Find a point c P which is visible from every point p P. Place a guard of color k on c. return Guard position and color. How can we find a star pocket? A star pocket is star-shaped, hence it can t have two consecutive reflex vertices on its boundary. (Since the polygon is orthogonal, the vertical edges at these two vertices can t be guarded by a single guard.) Candidates for making a cut are therefore all horizontal edges with two reflex vertices x, o. Algorithm 4.2 finds these pairs and returns the pairs in descending order of their y-coordinates. This can be used to cut off star pockets from the polygon, starting at the top of the polygon. Algorithm 4.2 FindCandidates(Polygon P) Require: A monotone orthogonal polygon P over a single horizontal edge Find all pairs (x, o) of two consecutive reflex vertices in P. Let L be an array containing all these pairs. Sort L by descending y-coordinate. return L 22 However we must be careful to cover only pieces that are independent with guards of the same color. Algorithm 4.3 on the next page achieves this by ensuring that no two star pockets on top of each other get a guard of the same color. The algorithm uses each round of a while-loop to cut independent

29 4.2. The building blocks star pockets, starting at the top of the polygon and proceeding downwards. The cut off regions in each round receive the same color. A cut is made at a candidate vertex in round i if and only if there hasn t been a cut above it in round i. Hence we have only guards above the cut with color smaller than i. Algorithm 4.3 GuardSingleEdge(Polygon P) Require: A monotone polygon P over a single horizontal edge {Set the initial color i, an initially empty guard set G and get star pocket candidates} i 1 G newarray() L FindCandidates(P) {While P is not a star-shaped polygon, cut independent star pockets and cover them with guards of color i.} while L is non-empty do {Maintain a list C of the currently cut subpolygons of color i, represented by a vertex of the cut-edge} C newarray() {Go through all candidates, starting at the top of the polygon.} for all (x, o) in L do {For both reflex vertices, cut off a star pocket if possible} for all v {x, o} do if A cut at vertex v has no cut in C above it then P 0 subpolygon of P above the cut. G += GuardStar(P 0, i) P P\P 0. C += v end if end for end for {Get new star pocket candidates for the updated P, use a new color.} L FindCandidates(P) i += 1 end while {The truncated polygon P is now star-shaped.} G += GuardStar(P, i) return G Figure 4.4 on the following page shows the cuts from the first two rounds of the algorithm. A pair (x, o) can give rise to either two cuts, only one cut or no cut at all. In the latter case, this pair is again contained in L in the next round of the while-loop. As we can see there is also a new pair in round two, that consists of two reflex vertices whose adjacent vertices in the first 23

30 4. Conflict-free guard coloring in orthogonal polygons Figure 4.4: The cuts in the first and second round of Algorithm 4.3. round are defining vertices of the same cut. Lemma 4.3 For every monotone orthogonal polygon P n on n vertices over a single horizontal edge as an input, GuardSingleEdge (Algorithm 4.3) terminates and gives a guard placement G with a conflict-free coloring using at most log n colors. Proof Note that any cut made by the algorithm gets rid of at least two vertices above the cut plus at least one vertex adjacent to the cut, while introducing at most one new vertex to the truncated polygon. Furthermore at least one cut is made in each round of the while-loop, hence the number of remaining unguarded vertices decreases by at least two in each round. Therefore the algorithm terminates. Each subpolygon that gets cut in the while-loop also gets guarded by Algorithm GuardStar. The same holds for the truncated polygon that remains after the while-loop. Hence G contains indeed a guard set. Furthermore the coloring is conflict-free: For every subpolygon guarded by a guard of color i the algorithm ensures that there is no other star pocket above that region that contains a guard of the same color. Such a subpolygon is also bounded to the left and to the right by edges of the polygon. Hence the star pockets Figure 4.5: Proof of the independence of the red star pockets. 24 (not the subpolygons!) are independent and a conflict can only arise on the bottom edge of a cut subpolygon. But this bottom edge is also contained in the boundary of the subpolygon below, see Figure 4.5. Therefore also the edge is guarded conflict-free by the guard of the lower star pocket. Hence two regions that are guarded by the same color are independent.

31 4.2. The building blocks Let k be the highest color given to any guard by GuardSingleEdge. To prove that we don t need more than log n colors we show that 2 k n. Look at a schematic graph T of the colored guards in G. The vertices of T correspond to the guards in G and two vertices are connected in T if the star pockets they guard are adjacent. Since the algorithm does only horizontal cuts, T is clearly a tree. We give an example of the algorithms result for a polygon P n and the corresponding tree T in Figure 4.6. Figure 4.6: A schematic tree of the guards given by Algorithm 4.3. We show the following property of T: every node s of color i (for 2 i k) has at least two children of color i 1. Assume for the sake of contradiction there exists a vertex s of color i that doesn t have two children of color i 1. Assume s has no children of color i 1 at all. Since the algorithm ensures that a cut subpolygon only has guards of lower color above it, the subtree T s T rooted at s has no color higher than i 2. But then the algorithm could have executed the cut of the subpolygon guarded by s already at least one loop before, which is a contradiction. Therefore s must have exactly one child of color i 1. Assume without loss of generality that the region guarded by that child was cut by a cut-line originating at a reflex vertex x. But then the algorithm should have made another cut-line originating at the adjacent vertex o (adjacent in round i 1), since there is no region of color i 1 above that cut-line by assumption. This is again a contradiction, hence every node s of color 2 i k has at least two children of color i 1. Every region guarded by a guard of color 1 has at least two vertices not guarded by another guard of color 1. Furthermore the tree has at least 2 k 1 leafs (i.e. guards of color 1) by the given property of T. Since P has n vertices of which none is guarded by more than one leaf guard, we get the desired inequality 2 2 k 1 = 2 k n and hence k log n. Using the building blocks In the next three sections we decrease our chromatic art gallery problem in its complexity. Namely, we partition P into four sets of regions and characterize the relations between two regions of the same set. We then use this to show how to guard the regions of each 25

32 4. Conflict-free guard coloring in orthogonal polygons set with at most a logarithmic number of colors to achieve a conflict-free guard coloring. To do this we need our results on staircase polygons and monotone polygons over a single horizontal edge. 4.3 Dividing into four equivalent subproblems The basic idea is to partition the polygon iteratively into four types of monotone orthogonal polygons which have a boundary consisting of a single base edge and another subchain. Our monotone polygons are either x-monotone or y-monotone, and they can lie either above the base edge or below in the former case, and to the left or to the right in the latter case. We use mnemonic identifiers U (up), D (down), L (left) and R (right) to refer to these four types. In the figures where we show all or parts of the partition, we display these types with the colors red, green, black and blue. Partitioning Given any polygon P we construct a partition by iteratively adding monotone subpolygons. In each step we either add subpolygons of Type U and D or subpolygons of Type L and R. Step 1 We start with the lowest horizontal edge of P s boundary. This edge determines our first subpolygon of Type U (see Figure 4.7) in the following way: Let l denote the lowest horizontal edge. Then the corresponding sub- l 1 Figure 4.7: The first step of partitioning an orthogonal polygon. polygon Q is the set of all points q P which are vertically visible from l and lie on or above l. The boundary of Q consists of edges in P and of vertical line segments with endpoints on P. 26

33 4.3. Dividing into four equivalent subproblems Remark 4.4 Since P is a simply connected region, P has no holes. This means that Q splits P in parts to its right and to its left. Such a part R has exactly one common edge with Q, either a vertical line segment on the left side of Q or a vertical line segment on the right side of Q (otherwise R Q contains a hole in its interior which is a contradiction to P being simply connected). Step 2 These line segments of Q are the base edges for subpolygons of Type L and R, which are defined analogously as the first subpolygon, with vertical visibility replaced by horizontal visibility. Again, these new subpolygons give horizontal line segments in P, see Figure 4.8. Remark 4.5 Reconsidering Remark 4.4 gives us that the remaining regions boundaries share exactly one line segment with subpolygons of Type L or Type R, but not with the first subpolygon Figure 4.8: The second step of partitioning an orthogonal polygon. Step 3 In the same way as in Step 2, the horizontal line segments from Step 2 give rise for subpolygons of Type U and D, see Figure 4.9 on the next page. We repeat steps 2 and 3 until we have a complete partition of the polygon, see Figure 4.11 on page 29. Proposition 4.6 The partitioning process gives a complete partition after a finite number of steps. Proof While the polygon is not completely covered by subpolygons of the partition, in each step at least one subpolygon is added to the partition. Such a subpolygon touches at least one edge e previously not touched by a subpolygon, see Figure In the next step at least one vertex v of that 27

34 4. Conflict-free guard coloring in orthogonal polygons Figure 4.9: The third step of partitioning an orthogonal polygon. edge is touched by a new subpolygon. After another step, that vertex is completely surrounded by subpolygons. Since in each step at least one such vertex surrounding is started and the polygon is completely covered if all vertices are completely surrounded, the partitioning process ends after at most n + 2 steps. e v Figure 4.10: In each consecutive three steps at least one new vertex is completely surrounded. Schematic tree Given a complete partition of a polygon P which has been constructed by this process, we can represent the partition by a schematic tree. The schematic tree T is a 4-colored directed graph, where each vertex represents a subpolygon of the partition of the same color. There exists a directed edge from a subpolygon P i to a subpolygon P j if and only if P j has been constructed over a line segment that is part of P i s boundary. Because of Remark 4.5, T contains no cycle. Hence the graph T is indeed a tree. The first constructed subpolygon is corresponding to the only vertex with no incoming edges. We fix this vertex as the root of the schematic tree. Since all other vertices have indegree 1, T is a rooted directed tree and any subpolygon constructed in Step k has depth k 1 in T. Hence all red and 28

35 4.4. A necessary condition for intersections Figure 4.11: The complete partition. green vertices have even height and all vertices of color blue or black have odd height. Therefore, every directed path alternates between vertices of red or green color and vertices of blue or black color. In Figure 4.12 we show the schematic tree representing the partition in Figure Step 6 Step 5 Step 4 Step 3 Step 2 Step 1 Figure 4.12: The schematic tree of the partition in Figure A necessary condition for intersections Each subpolygon of type U is easily guarded with a conflict-free coloring using O(log n) colors by Lemma 4.3, but complications arise in showing that all the subpolygons of type U can be colored within the same bound. We do this by establishing independence between almost all of these subpolygons. More precisely, we show independence for the subpolygons without their base edges. We consider the base edge l i of a subpolygon P i to be part of 29