CONSISTENT LABELING OF ROTATING MAPS

Transcription

1 CONSISTENT LABELING OF ROTATING MAPS Andreas Gemsa, Martin Nöenburg, Ignaz Rutter Abstract. Dynamic maps that aow continuous map rotations, for exampe, on mobie devices, encounter new geometric abeing issues unseen in static maps before. We study the foowing dynamic map abeing probem: The input is an abstract map consisting of a set P of points in the pane with attached horizontay aigned rectanguar abes. Whie the map with the point set P is rotated, a abes remain horizontay aigned. We are interested in a consistent abeing of P under rotation, i.e., an assignment of a singe (possiby empty) active interva of anges for each abe that determines its visibiity under rotations such that visibe abes neither intersect each other (soft conficts) nor occude points in P at any rotation ange (hard conficts). Our goa is to find a consistent abeing that maximizes the number of visibe abes integrated over a rotation anges. We first introduce a genera mode for abeing rotating maps and derive basic geometric properties of consistent soutions. We show NP-hardness of the above optimization probem even for unit-square abes. We then present a constant-factor approximation for this probem based on ine stabbing, and refine it further into an efficient poynomia-time approximation scheme (EPTAS). 1 Introduction Dynamic maps, in which the user can navigate continuousy through space, are becoming increasingy important in scientific and commercia GIS appications as we as in persona mapping appications. In particuar, GPS-equipped mobie devices offer various new possibiities for interactive, ocation-aware maps. A common principe in dynamic maps is that users can pan, rotate, and zoom the map view. Despite the popuarity of severa commercia and free appications, reativey itte attention has been paid to provaby good abeing agorithms for dynamic maps. Been et a. [2] identified a set of consistency desiderata for dynamic map abeing. Labes shoud neither jump (suddeny change position or size) nor pop (appear and disappear more than once) during monotonous map navigation; moreover, the abeing shoud be a function of the seected map viewport and not depend on the user s navigation history. Previous work on the topic has focused soey on supporting zooming and/or panning of the A preiminary version of this paper was presented at the 12th Internationa Symposium on Agorithms and Data Structures (WADS 2011) [7]. Agorithms and Compexity Group, TU Wien, Vienna, Austria, noeenburg@ac.tuwien.ac.at Institute of Theoretica Informatics, Karsruhe Institute of Technoogy (KIT), Karsruhe, Germany, rutter@kit.edu JoCG 7(1), ,

2 (a) Labe 5 Labe 4 Labe 3 Labe 2 Labe 1 (b) Labe 5 Labe 4 Labe Labe 3 Labe 2 1 (c) Labe 4 Labe 1 Labe 5Labe 2 Labe 3 (d) Labe 1 Labe 2 Labe 4 Labe 3 Labe 5 Figure 1: Input map with five points (a) and three rotated views with some partiay occuded abes (b) (d). map [2, 3, 15, 19], whereas consistent abeing under map rotations has not been considered prior to the conference version of this paper [7]. Most maps come with a natura defaut orientation (usuay the northern direction facing upward), but appications such as car or pedestrian navigation often rotate the map view dynamicay to be aways forward facing [10]. Sti, the abes, usuay modeed as rectanges, must remain horizontay aigned for best readabiity regardess of the actua rotation ange of the map. A basic requirement in static and dynamic abe pacement is that abes are pairwise disjoint and thus form an independent set of rectanges, which means that in genera not a abes can be paced simutaneousy. For abeing point features unambiguousy, it is further required that each abe rectange contains the abeed point (aso caed anchor point), either in its inside or on its boundary. Further, we do not aow that abes occude other input points, even if the points are unabeed at a particuar rotation ange. Figure 1 shows a sketch of a map that is rotated and abeed, depending on the rotation ange. The objective in map abeing is usuay to pace as many abes as possibe. Transating this into the context of rotating maps means that, integrated over one fu rotation from 0 to 2π, we want to maximize the number of visibe abes. The consistency requirements of Been et a. [2] can immediatey be appied to rotating maps. Our Resuts. Initiay, we define a mode for rotating maps and show some basic properties of the different types of conficts that may arise during rotation. Next, we prove that consistenty abeing rotating maps is NP-hard, both for the maximization of the tota number of visibe abes integrated over one fu rotation and for the maximization of the smaest visibiity range of any abe. Under the assumption of a bounded feature density in the input map we present a 1/4-approximation agorithm and an efficient poynomia-time approximation scheme (EPTAS) for unit-height rectanges 1. We extend both agorithms to the case of rectanguar abes with the property that the ratio of the smaest and argest width, the ratio of the smaest and argest height, as we as the aspect ratio of every abe is bounded by a constant, even if we aow the anchor point of each abe to be an arbitrary point of the abe. This appies to most practica scenarios, where abes typicay consist of few and reativey short ines of text. Moreover, our agorithmic resuts aso appy to the extensions of the abeing mode for rotating maps introduced in our foow-up paper [9]. 1 A PTAS is caed efficient if its running time is O(f(ε) n c ) for some constant c independent of ε. JoCG 7(1), ,

3 Reated Work. Most previous agorithmic research efforts on automated abe pacement cover static abeing modes for point, ine, or area features. For static point abeing, fixed-position modes and sider modes have been introduced [5, 21], in which the abe, represented by its bounding box, needs to touch the abeed point with one of its corners or aong its boundary. The abe number maximization probem is NP-hard even for the simpest abeing modes [12, 17, 21], whereas there are efficient agorithms for the decision probem that asks whether a points can be abeed in some of the simper modes. This decision probem is NP-hard in the fixed-position mode with four different positions, but is in P if ony two positions per abe are aowed [5]. Approximation resuts [1, 4, 21], heuristics [23], and exact approaches [13, 22] are known for many variants of the static abe number maximization probem. In addition to the abe number maximization probem, there is aso the abe size maximization probem, where a abes need to be paced and the optimization goa is to find the argest factor by which a abes can be scaed such that not two abes overap each other. As for the abe size maximization probem, approximation agorithms are known [5, 23]. In recent years, dynamic map abeing has emerged as a new research topic that gives rise to many unsoved agorithmic probems. Petzod et a. [20] used a preprocessing step to generate a reactive confict graph that represents possibe abe overaps for maps of a scaes. For any fixed scae and map region, their method computes a confict-free abeing using static abeing heuristics. Mote [18] presents another fast heuristic method for dynamic confict resoution in abe pacement that does not require preprocessing. The consistency desiderata of Been et a. [2] for dynamic abeing (no popping and jumping effects during panning and zooming), however, are not satisfied by either of the methods as they were mainy designed for quicky abeing a static map of arbitrary scae. Been et a. [3] showed NP-hardness of the abe number maximization probem in the consistent abeing mode and presented severa approximation agorithms. Gemsa et a. [8] considered a sider mode and presented an FPTAS for abeing a one-dimensiona zoomabe point set, where for each abe not ony a consistent visibiity range but aso an optima sider position for a zoom eves needs to be found. Nöenburg et a. [19] studied a dynamic version of the aternative boundary abeing mode, in which abes are paced at the sides of the map and connected to their points by eaders. They presented an agorithm to precompute a data structure that represents an optima one-sided abeing for a possibe scaes and thus aows continuous zooming and panning. None of the existing dynamic map abeing approaches supports map rotation. After the pubication of the conference version of this paper, rotating maps have been further studied. Gemsa et a. [6] discussed the probem of abeing a rotating and transating map, where ony a sma part of the map can be seen on screen. This is a probem that manufacturers of modern GPS navigation devices face when impementing the agorithms for dispaying the map. They presented an NP-hardness proof as we as an approximation agorithm for unit-square abes. The restricted case that no more than k abes may be shown simutaneousy can be soved efficienty. Yokosuka and Imai [24] investigated the abe size maximization probem for rotating maps. They presented efficient agorithms for finding anchor points inside or on the boundary of the abes such that the abe size is maximized and no conficts occur during a fu rotation. Finay, in a foow-up paper [9], JoCG 7(1), ,

4 we have recenty experimentay evauated severa abeing strategies for rotating maps and compared the achievabe activity eves for different degrees of consistency in the sense of Been et a. [2]. Our resuts confirmed that the mode we propose in this paper offers a good trade-off between consistency and abe activity. We further evauated the performance of the 1/4-approximation agorithms proposed in this paper in comparison to additiona heuristics and an exact ILP approach. 2 Mode In this section we describe our abeing mode for rotating maps with axis-aigned rectanguar abes. Athough this paper focuses on a theoretica understanding of one particuar mode, the one introduced in the conference version of this paper, our presentation is inspired by the more genera famiy of modes from the foow-up paper [9]. Let M be an (abstract) map, consisting of a set P = {p 1,..., p n } of points in the pane together with a set L = { 1,..., n } of cosed, axis-aigned, and not necessariy disjoint rectanguar abes in the pane. Each point p i must ie inside (or on the boundary of) its corresponding abe i at an arbitrary but fixed position; we denote p i as the anchor of abe i and i as anchored at p i. Note that we do not aow movement of a abe reative to its anchor, i.e., i must maintain the same position reative to p i. As M rotates, each abe i in L must remain horizontay aigned and anchored at p i. Thus, new abe intersections form and existing ones disappear during the rotation of M. We take the foowing aternative perspective on the rotation of M. Rather than rotating the point set P, say cockwise, and keeping the abes horizontay aigned, we may instead rotate each abe countercockwise around its anchor point and keep the set of points fixed. It is easy to see that both rotations are equivaent in the sense that they yied exacty the same intersections of abes and occusions of points. We consider a rotation anges moduo 2π. For convenience we introduce the interva notation (a, b) for any two anges a, b [0, 2π]. If a b, this corresponds to the standard meaning of an open interva, otherwise, if a > b, we define (a, b) := (a, 2π] [0, b). For simpicity, we refer to any set of the form (a, b) as an interva. We further define the ength of an interva I as I = b a if I = (a, b) with a b and I = 2π a + b if I = (a, b) with a > b. We further extend the above definitions to the standard interva notation for haf-open and cosed intervas of the form (a, b], [a, b), and [a, b]. A rotation of L is defined by a rotation ange α [0, 2π). We define the rotated abe set L(α) of a abes, each rotated by an ange of α around its anchor point. A rotation abeing of M is a function φ: L [0, 2π) {0, 1} such that φ(, α) = 1 if abe is visibe or active in the rotation of L by α, and φ(, α) = 0 otherwise. We ca a abeing φ vaid if, for any rotation α, the set of abes L φ (α) = { L(α) φ(, α) = 1} consists of pairwise disjoint abes. If two abes and in L(α) intersect, we say that they have a soft confict (or a abe-abe confict) at α, i.e., in a vaid abeing at most one of them can be active at α. We define the set C(, ) = {α [0, 2π) and are in confict at α} as the confict set of and. Further, we ca the begin and end of a maxima contiguous confict range in C(, ) confict or abe events. JoCG 7(1), ,

5 For a abe we ca each maxima interva I [0, 2π) with φ(, α) = 1 for a α I an active range of abe and define the set A φ () as the set of a active ranges of in φ. We ca an active range where both boundaries are confict events a reguar active range. Our optimization goa is to find a vaid abeing φ that maximizes the number of active abes integrated over one fu rotation from 0 to 2π. The vaue of this integra is caed the tota activity t(φ) and can be computed as t(φ) = L I A φ () I. A vaid abeing is not yet consistent in terms of the definition of Been et a. [2, 3]: for given fixed anchor points, abes ceary do not jump and the abeing is independent of the rotation history, but abes may sti ficker mutipe times during a fu rotation from 0 to 2π, depending on how many active ranges they have in φ. To reduce fickering effects, we aow each abe to have ony a singe active range. Accordingy, we ca such a abeing consistent. Since a consistent abeing aows ony one active range, this is caed the 1R-mode. We appy another restriction to our consistency mode, which is based on the occusion of anchors. Among the conficts in set C(, ) we further distinguish hard conficts (or abe-point conficts), i.e., conficts where abe intersects the anchor of abe. In our definition every hard confict is aso a soft confict as ceary the two abes intersect as we. If a abeing φ sets active during a hard confict with, the anchor of is occuded. This may be undesirabe in some situations in practice, for exampe, if every point in P carries usefu information in the map, even if it is currenty unabeed. Thus we may optionay require that φ(, α) = 0 during any hard confict of a abe with another abe at ange α. In that sense, the occuded anchor dictates that the occuding abe cannot be active. The objective in static map abeing is usuay to find a maximum subset of pairwise disjoint abes, i. e., to abe as many points as possibe. Generaizing this objective to rotating maps means that integrated over a rotations α [0, 2π) we want to dispay as many abes as possibe. This corresponds to finding a vaid rotation abeing φ with maximum tota activity t(φ) over a vaid rotation abeings; we ca this optimization probem MaxTota. An aternative objective is to maximize over a vaid rotation abeings φ the minimum ength min{ I A φ () L} of a active ranges; this probem is caed MaxMin. We note that the 1R-mode readiy generaizes to the kr-mode that aows at most k active ranges per abe. The version without any consistency constraints is caed Rmode; it mainy serves as an upper bound for the other modes. Finay, a more drastic approach that eiminates a fickering is the 0/1-mode, where every abe is either active for the fu rotation [0, 2π) or never at a. Our experimenta evauation [9] showed that the 0/1-mode is too restrictive as it often ony achieves 50% or ess of the abe activity of an optima soution in the R-mode, whereas an optima soution for the 1R-mode achieves between 80% and 95% of the maximum activity in the R-mode for severa reaword instances. Hence the 1R-mode is a reasonabe choice that offers good soutions whie containing sufficient structure for a theoretica treatment. For this paper we consider the 1R hard-confict mode ony; whenever we use the term optima rotation abeing, this refers to a vaid optima rotation abeing in this particuar mode. We first give some compexity resuts and then deveop approximation agorithms, most notaby an EPTAS for the case that the rectanguar abes have bounded JoCG 7(1), ,

6 r t r b (a) r b t t r b b r t r r t b (b) b t b t t r b b r t r b t r b t (c) b t t t b r b r b r b r t t (d) r b t t r b r (e) r (f) t b (g) t b (h) r Figure 2: Two abes and and their eight possibe boundary intersection events. Anchor points are marked as back dots. width ratio, bounded height ratio and bounded aspect ratio. It shoud be noted that, whie our hardness resuts rey on both the hard conficts and the restriction to a singe active range, our agorithmic resuts do not have this imitation and generaize to the kr-mode with or without expicit hard conficts. 3 Properties of Rotation Labeings In this section we show basic properties of rotation abeings. If two abes and intersect in a rotation of α, they have a confict at α, i.e., in a consistent abeing at most one of them can be active at α. Lemma 1. For any two abes and with anchor points p and p the set C(, ) consists of at most four disjoint contiguous confict ranges. Proof. The first observation is that due to the simutaneous rotation of a initiay axisparae abes in L, and remain parae at any rotation ange α. Rotation is a continuous movement and hence any maxima contiguous confict range in C(, ) must be a cosed interva [α, β], where 0 α, β 2π. At a rotation of α (resp. β) the two abes and intersect ony on their boundary. Let, r, t, b be the eft, right, top, and bottom sides of and et, r, t, b be the eft, right, top, and bottom sides of (defined at a rotation of 0). Since and are parae, the ony possibe cases in which they intersect on their boundary but not in their interior are t b, b t, r, and r. Each of those four cases may appear twice, once for each pair of opposite corners contained in the intersection. Figure 2 shows a eight boundary intersection events. Each of the conficts defines a unique rotation ange and obviousy at most four disjoint confict ranges can be defined with these eight rotation anges as their endpoints. In the foowing we ook more cosey at the conditions under which the boundary intersection events (the confict events) occur and at the rotation anges defining them. Let h t and h b be the distances from p to t and b, respectivey. Simiary, et w and w r be the distances from p to and r, respectivey; see Figure 3. By h t, h b, w, and w r we JoCG 7(1), ,

7 h t w h b t w r p b Figure 3: Parameters of abe anchored at p. r p t α d p b h t + h b (a) rotation of 2π α p d t h t + h b (b) rotation of π + α b α p Figure 4: Boundary intersection events for t b. denote the corresponding vaues for abe. Finay, et d be the distance of the two anchor points p and p. To improve readabiity of the foowing emmas we define two functions f d (x) = arcsin(x/d) and g d (x) = arccos(x/d). Lemma 2. Let and be two abes anchored at points p and p that ie on a horizonta ine. Then the confict events in C(, ) are a subset of C = {2π f d (h t + h b ), π + f d(h t + h b ), f d(h b +h t), π f d (h b +h t), 2π g d (w r +w ), g d(w r +w ), π g d(w +w r), π+g d (w +w r)}. Proof. First we show that the possibe confict events are precisey the rotation anges in C. We start considering the intersection of the two sides t and b. If there is a rotation ange under which t and b intersect then we have the situation depicted in Figure 4 and by simpe trigonometric reasoning the two rotation anges at which the confict events occur are 2π arcsin((h t +h b )/d) and π+arcsin((h t+h b )/d). Obviousy, we need d h t+h b. Furthermore, for the intersection in Figure 4a to be non-empty, we need d 2 (w r + w )2 + (h t + h b )2 ; simiary, for the intersection in Figure 4b, we need d 2 (w + w r) 2 + (h t + h b )2. From an anaogous argument we obtain that the rotation anges under which b and t intersect are arcsin((h b + h t)/d) and π arcsin((h b + h t)/d). Ceary, we need d h b + h t. Furthermore, we need d 2 (w r + w )2 + (h b + h t) 2 for the first intersection and d 2 (w + w r) 2 + (h b + h t) 2 for the second intersection to be non-empty under the above rotations. The next case is the intersection of the two sides r and, depicted in Figure 5. Here the two rotation anges at which the confict events occur are 2π arccos((w r + w )/d) and arccos((w r + w )/d). For the first confict event we need d2 (w r + w )2 + (h t + h b )2, and for the second we need d 2 (w r + w )2 + (h b + h t) 2. For each of the intersections to be non-empty we additionay require that d w r + w. Simiar reasoning for the fina confict events of r yieds the rotation anges π arccos((w + w r)/d) and π + arccos((w + w r)/d). The additiona constraints are d w + w r for both events and d 2 (w + w r) 2 + (h b + h t) 2 for the first intersection and JoCG 7(1), ,

8 p β w r + w r d p w r + w p β d r p (a) rotation of 2π β (b) rotation of β Figure 5: Boundary intersection events for r. hard confict range p p Figure 6: Confict ranges of two abes and marked in bod on the encosing circes. d 2 (w +w r) 2 +(h t +h b )2 for the second intersection. Thus, C contains a possibe confict events. One of the requirements for a vaid abeing in the hard-confict mode is that no abe may contain a point in P other than its anchor point. For each abe this gives rise to the specia cass of hard confict ranges, in which may never be active. We note that by definition soft conficts are symmetric, i.e., C(, ) = C(, ), whereas hard conficts are not symmetric. The next emma characterizes the hard confict ranges. Lemma 3. For a abe anchored at point p and a point q p in P such that p and q ie on a horizonta ine, the hard confict events of and q are a subset of H = {2π f d (h t ), π + f d (h t ), f d (h b ), π f d (h b ), 2π g d (w r ), g d (w r ), π g d (w ), π + g d (w )}. Proof. We define a abe of width and height 0 for q, i.e., we set h t = h b = w = w r = 0. Then the resut foows immediatey from Lemma 2. Obviousy, Lemma 2 and Lemma 3 can aso be used to compute the confict events when the respective points do not ie on a horizonta ine. In this case, we simpy rotate the instance by an ange α such that the (anchor) points are horizontay aigned, then we compute the set of confict events according to the emma, and finay we shift the computed anges of confict events by α to offset the initia rotation. A simpe way to visuaize confict ranges and hard confict ranges is to mark, for each abe anchored at p and each of its (hard) confict ranges, its corresponding circuar arc on the circe centered at p and encosing. Figure 6 shows an exampe. In the foowing we show that the MaxTota probem can be discretized in the sense that there exists an optima soution whose active ranges are defined as intervas JoCG 7(1), ,

9 whose borders are abe events. An active range border of a abe is an ange α that is characterized by the property that the abeing φ is not constant in any ε-neighborhood of α. Reca that a reguar active range is an active range whose borders are both abe events. Lemma 4 (Discretization Lemma). Given a abeed map M there is an optima rotation abeing of M consisting of ony reguar active ranges. Proof. Let φ be an optima abeing with a minimum number of active range borders that are not abe events. Assume that there is at east one active range border β that is not a abe event. Let α and γ be the two adjacent active range borders of β, i.e., α < β < γ, where α and γ are active range borders, but not necessariy abe events. Then et L be the set of abes whose active ranges have eft border β and et L r be the set of abes whose active ranges have right border β. For φ to be optima L and L r must have the same cardinaity since otherwise we coud increase the active ranges of the arger set and decrease the active ranges of the smaer set by an ε > 0 and obtain a better abeing. So define a new abeing φ that is equa to φ except for the abes in L and L r : define the eft border of the active ranges of a abes in L and the right border of the active ranges of a abes in L r as γ instead of β. Since L = L r we shrink and grow an equa number of active ranges by the same amount. Thus the two abeings φ and φ I A φ () I = L I A φ () I. Because φ uses as have the same objective vaue L active range borders one non-abe event ess than φ this number was not minimum in φ a contradiction. As a consequence φ has ony abe events as active range borders. 4 Compexity Considerations In this section we show that finding an optima soution for MaxTota (and aso MaxMin) is NP-hard in the 1R hard-confict mode even if a abes are unit squares and their anchor points are their ower-eft corners. We present a gadget proof reducing from the NP-compete probem Panar 3Sat [16], which is the restriction of 3Sat for panar formuas defined as foows. Let ϕ be a Booean 3Sat formua and et G ϕ denote the variabe cause graph, which contains a vertex v x for each variabe x of ϕ and a vertex v C for each cause C of ϕ such that v x and v C are adjacent if and ony if C contains x or x. The formua ϕ is panar if and ony if G ϕ is panar. Every panar variabe cause graph G ϕ admits a panar grid ayout where a vertices are positioned on the x-axis and the causes are drawn as three-egged combs above and beow the x-axis [14]; see Fig. 7. The ayout fits on a poynomiay bounded integer grid. The genera idea behind the reduction from an instance ϕ of Panar 3Sat is to construct an instance I ϕ of MaxTota whose geometric ayout mimics the panar grid ayout of G ϕ and whose optima rotation abeings are reated to the satisfiabiity of ϕ in the foowing sense. There is a vaue K depending ony on ϕ such that ϕ is satisfiabe if and ony if I ϕ admits a rotation abeing with tota activity at east K. For this reduction, we describe how to buid gadgets that correspond to the variabes and causes of the Panar 3Sat formua, caed variabe and cause gadgets, respectivey, JoCG 7(1), ,

10 x 1 x 3 x 5 x 1 x 2 x 3 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 x 2 x 3 x 4 x 1 x 2 x 4 x 1 x 4 x 5 Figure 7: An instance of Panar 3Sat with a grid ayout of the variabe cause graph. as we as pipes that mode the edges of G ϕ. The fact that our ayout mimics the panar grid ayout of G ϕ ensures that the ony pairs of abes that interact via conficts are either part of the same gadget or they are in adjacent gadgets, i.e., one is in a variabe gadget and one is in a pipe or one is in a pipe and the other one is in a cause gadget. Since the gadgets need to propery correspond to Booean variabes, the gadgets can have, in an optima MaxTota soution, ony two possibe states; one which corresponds to true, and the other to fase. Simiary, the cause gadgets shoud contribute the same vaue to the tota activity of the optima soution if and ony if at east one of the cause s iteras is in the state true; otherwise, if a three iteras are fase, a cause gadget contributes a smaer vaue. Obviousy, we need aso some kind of connection between the variabe gadgets and the cause gadgets to transmit the states of the variabe gadgets. Those shoud contribute to the vaue of the optima soution of MaxTota the same vaue regardess of the state of the connected variabe. If this is the case, then we ony need to choose the appropriate vaue K, for which the vaue of the optima soution of the instance indicates that the corresponding Panar 3Sat formua has a satisfying truth assignment. Since the construction of the MaxTota instance can be carried out in poynomia time, it foows that MaxTota is NP-hard. The same gadgets can aso be used to prove NP-hardness of MaxMin as we wi see in the end of this section. 4.1 Basic Buiding Bocks We start by constructing basic buiding bocks, caed chain, inverter, and turn, from which we construct our gadgets. As a preiminary step we need a specia property of unit-square abes. Lemma 5. If two unit-square abes and whose anchor points are their ower-eft corners have a confict at a rotation ange α, then they have conficts at a anges α + i π/2 for i Z. Proof. Simiar to the notation used in Section 3, et f d = arcsin(1/d) and g d = arccos(1/d). From Lemma 2 we obtain the set C = {2π f d, π + f d, f d, π f d, 2π g d, g d, π g d, π + g d } JoCG 7(1), ,

11 of confict events for which it is necessary that the distance d between the two anchor points is 1 d 2. Since arccos x = π/2 arcsin x the set C can be rewritten as C = {f d, π/2 f d, π/2 + f d, π f d, π + f d, 3π/2 f d, 3π/2 + f d, 2π f d }. This shows that conficts repeat after every rotation of π/2. For every abe we define the outer circe of as the circe of radius 2 centered at the anchor point of. Since the top-right corner of traces the outer circe we wi use the ocus of that corner to visuaize active ranges or confict ranges on the outer circe. Note that due to the fact that at the initia rotation of 0 the diagona from the anchor point to the top-right corner of forms an ange of π/4 a marked ranges are actuay offset by π/4. We now describe the buiding bocks and prove their basic properties. Chain. A chain consists of at east four abes anchored at coinear points that are eveny spaced with distance 2. Hence, each point is paced on the outer circes of its neighbors. We ca the first two and ast two abes of a chain terminas and the remaining part inner chain, see Figure 8. A chain is horizonta (vertica) if its anchor points are horizontay (verticay) aigned. We denote an assignment of active ranges to the abes as the state of the chain. The important observation is that in any optima soution of MaxTota (and MaxMin) an inner chain has ony two different states, whereas terminas have mutipe optima states that are a equivaent for the purpose of MaxTota; see Figure 8. In particuar, in an optima soution each abe of an inner chain has an active range of ength π and active ranges aternate between adjacent abes. We wi use the two states of chains as a way to encode truth vaues in our reduction. In an optima soution of MaxTota, each termina contributes 5π/2 to the objective function. One way to achieve this is by assigning active ranges of ength π to the inner termina abes and active ranges of 3π/2 to the outer termina abes, see the eft termina in Figure 8. Another possibiity is to assign an active range of π/2 to the inner termina abe (starting at the ange 0, π/2, π, or 3π/2) and 2π to its neighboring outer termina abe, see the right termina in Figure 8. For MaxTota a states behave equivaenty, for MaxMin we observe that the first termina configuration with minimum active range π is aways possibe. termina inner chain termina Figure 8: Chain. It has ony two states in any optima soution of MaxTota. One of the states is marked by the soid bue arcs, whie the other is marked by the dotted red arcs. The soid back arcs of the right termina show an aternative configuration that is independent of the state of the inner chain. JoCG 7(1), ,

12 6/2 inner chain p a p b p c inner chain/termina 2/2 p d inner chain/termina Figure 9: Turn. A turn that spits one inner chain into two inner chains. Like the chain, the turn has ony two possibe states in an optima soution. One is marked by the soid bue arc, whie the other is marked by the red dotted arc. Connecting gadgets are indicated by abeed arrows. Lemma 6. In any optima soution, any abe of an inner chain has an active range of ength π. The active ranges of consecutive abes of an inner chain aternate between (0, π) and (π, 2π). Proof. By construction every abe of an inner chain has two hard conficts at anges 0 and π, so no active range can have ength arger than π. From Lemma 5 we know that every abe of an inner chain further has conficts at π/2 and 3π/2. These conficts are soft conficts and can be resoved by either assigning a odd abes the active range (0, π) and a even abes the active range (π, 2π) or vice versa. Obviousy both assignments are optima and there is no optima assignment in which two adjacent abes have active ranges on the same side of π. For inner chains whose distance between two adjacent points is ess than 2 the ength of the confict region changes, but the above arguments remain vaid for any distance between 1 and 2. To ensure that each abe has a maxima active range of at east 2π/3, we need at east distance 1/ cos(π/12) = This impies that chains are compressibe in the sense that we can move some of their anchors coser together without vioating the overa behavior. We wi use this freedom ater to ensure that some anchors of our constructions end up at specific positions. Turn. The third buiding bock is a turn that consists of four abes; see Figure 9. The anchor points p a and p b are at distance 2 and the pairwise distances between p b, p c, and p d are aso 2 such that the whoe structure is symmetric with respect to the ine through p a and p b. The centra point p b is caed turn point, and the two points p c and p d are caed outgoing points. Due to the hard conficts created by the four points we observe that the outer circe of p b is divided into two ranges of ength 5π/6 and one range of ength π/3. The outer circes of the outgoing points are divided into ranges of ength π, 2π/3, and π/3. The outer circe of p a is divided into two ranges of ength π. The outgoing points serve as connectors to terminas, inner chains, or further turns. By couping mutipe turns we can divert an inner chain by any mutipe of 30. JoCG 7(1), ,

13 Lemma 7. A turn has ony two optima states and aows an inner chain to be spit into two equivaent parts in an optima soution. Proof. We show that the activity in an optima soution for the turn is 21π/6 and that there are ony two different active range assignments that yied this soution. Note that for the abe a the ength of its active range is at most π. For b it is at most 5π/6 and for c and d it is at most π. We first observe that c and d cannot both have an active range of ength π since by Lemma 5 they have a soft confict in the intersection of their ength-π ranges. Thus at most one of them has an active range of ength π and the other has an active range of ength at most 5π/6. But in that case the same argumentation shows that the active range of b is at most π/2. Combined with an active range of ength π for a this yieds in tota a sum of 20π/6. On the other hand, if one of c and d is assigned an active range of ength 2π/3 and the other an active range of ength π as indicated in Figure 9, the soft confict of b in one of its ranges of ength 5π/6 is resoved and b can be assigned an active range of maximum ength. This aso hods for a resuting in a tota sum of 21π/6. Since the gadget is symmetric there are ony two states that produce an optima soution for the engths of the active ranges. By attaching inner chains to the two outgoing points the truth state of the inner chain to the eft is transferred into both chains on the right. 4.2 Gadgets of the Reduction In the foowing, we construct the variabe, cause, and pipe gadgets. As mentioned before our reduction mimics the panar grid ayout of G ϕ (reca Figure 7). However, as we have seen (reca Figures 8 and 9), the distances between anchors of our constructions invove distances such as 2 and 6. Nonetheess, we construct our gadgets in such a way that the coordinates of a anchor points can be expressed in the number fied Q( 2, 3). This ensures two crucia properties for the NP-hardness reduction. Firsty, this aows to encode the anchor coordinates using a poynomia number of bits, and thus ensures that the reduction can be carried out in poynomia time. Secondy, a our constructions are exact in the sense that the coordinates of anchors shared by two gadgets can be expressed reative to an arbitrary reference point in each gadget. Variabe Gadget. The variabe gadget consists of an aternating sequence of two buiding bocks: horizonta chains and itera readers. A itera reader is a structure that aows us to spit the truth vaue of a variabe into one part running via a pipe towards a cause and the part that continues the variabe gadget; see Figure 10. The itera reader consists of four turns and three terminas extended with short inner chains on both sides. The first turn connects to a pipe and the other three are dummy turns needed to ead the variabe gadget back to the horizonta ine. By verticay mirroring the itera reader we can aso connect to pipes beow the variabe gadget. JoCG 7(1), ,

14 termina inner chain turn pipe turn turn turn inner chain termina termina Figure 10: Sketch of a itera reader where connecting gadgets are indicated by abeed arrows. Anchor points at odd (even) positions are marked by back (white) disks. In order to encode truth vaues we define the state in which the eftmost abe (and any other abe at an odd position) of the eftmost horizonta chain has active range (0, π) as true and the state with active range (π, 2π) as fase. A horizonta chains and itera readers of the variabe gadget have an even number of abes so that the truth vaues propagate consistenty through the variabe gadget with respect to the parity of the abe positions. Cause Gadget. The cause gadget consists of one inner and three outer abes, where the anchor points of the outer abes spit the outer circe of the inner abe into three equa parts of ength 2π/3; see Figures 11 and 12. Each outer abe further connects to a pipe and a termina. These two connector abes are paced so that the outer circe of the outer abe is spit into two ranges of ength 3π/4 and one range of ength π/2. The terminas can be paced such that they do not have a confict with the other pipes or terminas; see Figure 11b. The genera idea behind the cause gadget is as foows. The inner abe obviousy cannot have an active range arger than 2π/3. Each outer abe is paced in such a way that if it carries the vaue fase it has a soft confict with the inner abe in one of the three possibe active ranges of ength 2π/3; see Figure 12a for an iustration of this (for simpicity ony the eft outer abe and the inner abe are depicted). In this exampe the pipe coming from the eft side transmits the truth vaue fase. This forces the outer abe a with anchor p a to have the active range indicated by the red dotted arc. This spits the arc from p a to p c of the inner abe s outer circe into two arcs where one arc has ength π/2 and the other π/6. The functionaity of the other outer abes is identica. Hence, if a three pipes transmit the vaue fase then every possibe active range of the inner abe of ength 2π/3 is affected by a soft confict. Consequenty, its active range can be at most π/2. Aternativey, an outer abe s active range might get spit to aow the inner abe to have an active range with ength 2π/3, but then this outer abe s active range in turn can have ength at most π/2 instead of 3π/4. It is aso possibe that the active range of a abe in the pipe (incuding the turns), or in a variabe might get spit, but this aso woud again reduce JoCG 7(1), ,

15 termina pipe termina pipe pipe p c termina pipe p c termina p a p t1 p b p a termina p b pipe termina { p t2 } pipe (a) Cause gadget with one inner abe (hatched disk) and three outer abes with their anchors p a, p b, and p c paced on the boundary of the inner abe such that the intersect at the center of the hatched disk. (b) Same cause gadget as in a, but with a termina attached to the eft outer abe (consisting of two abes with anchors p t1 and p t2 ), and a pipe attached to the ower outer abe (anchors at the center of the soid grey circes). Figure 11: Iustration of the cause gadget. p c p c p c p c p a p a p a p a p b p b p b p b (a) Soft Confict. (b) Hard Confict. (c) Hard Confict. (d) Hard Confict. Figure 12: Iustration of the four conficts of the eft outer abe with the inner abe. The ony soft confict is depicted in a. A remaining conficts are hard conficts. the active range of that particuar abe to π/2. Note that there is an optima soution of any satisfiabe instance, in which any abe has an active range of at east 2π/3. On the other hand, if at east one of the pipes transmits true, the inner abe can be assigned an active range of ength 2π/3. As can be seen in the Figures 12b and 12c, when the outer abe a with anchor p a has the active range indicated by the soid bue ine (i. e., the attached pipe transmits the truth vaue true), no arc of the inner abe is affected. The inner abe can have an active range that corresponds to the arc from p c to p a of ength 2π/3. Lemma 8. There must be a abe in a cause or in one of the connecting pipes with an active range of ength at most π/2 if and ony if a three iteras of that cause evauate to fase. Proof. The active range for the ower-right outer abe that is equa to the state fase is (3π/4, 3π/2). For the two other outer abes the active range corresponding to fase is rotated by ±2/3π. Note that the outer cause abes can have an active range of at most JoCG 7(1), ,

16 3/4π and the inner cause abe can have an active range of at most 2/3π. For every itera that is fase one of the possibe active ranges of the inner cause abe is spit by a confict into two parts of ength π/2 and π/6. This confict is either resoved by assigning an active range of ength π/2 to the inner cause abe or by propagating the confict into the pipe or variabe where it is eventuay resoved by assigning some active range with ength at most π/2. Otherwise, if at east one pipe transmits true, the inner abe of the cause can be active for 2π/3 whie the outer cause abes have an active range of ength 3π/4 and no chain or turn has a abe that is visibe for ess than 2π/3. Pipes. Pipes propagate truth vaues of variabe gadgets to cause gadgets. We use three different types of pipes, which we ca eft pipe, midde pipe, and right pipe, depending on where the pipe attaches to the cause. A pipe is formed by a sequence of turns and chains. Note that since each turn changes the direction by 30, we can route the pipes at any ange that is an integer mutipe of 30. The construction is iustrated in Fig 13. One end of each pipe attaches to a variabe at the open outgoing abe of a itera reader (marked by pipe in Figure 10). Initiay, the pipe eaves the variabe gadget at an ange of 30 with respect to the positive x-axis. We first use two eft turns so that the ange is 90. We then attach a vertica chain to reach the target cause. For a eft or right pipe we further extend the construction by a sequence of three right turns (for a eft pipe) or a sequence of three eft turns (for a right pipe); see Fig 13b. Finay, for eft and right pipe, we attach one further turns at the end so that a eft pipe enters the cause gadget at an ange of 330, a midde pipe at an ange of 90, and a right pipe at an ange of 210 with respect to the positive x-axis. For causes beow the variabes the pipes are mirrored. We position our pipes such that they attach to a itera reader that represents the state of the variabe. For transmitting the state of a variabe gadget, we use a pipe with an even number of abes aong its main path, to transmit its negation, we use a pipe with an odd number of abes on the main path. Due to the aternation property of chains (Lemma 6) this ensure that the correct state is transmitted. Note that the horizonta position of the cause is determined by the position of the midde pipe. The engths of the vertica chains of a pipes and the engths of the horizonta chains of the eft and right pipe can be chosen such that we end up sufficienty cose to the position of its cause, e.g., by enarging chains by pairs of anchors so that a pipe gets onger without changing its parity. In this way we can ensure that the ast anchor, which attaches to the cause, has distance at most 2 2 from the target cause in both x- and y-direction. We then compress the horizonta and vertica chains by a sma amount to reach the exact target position. 4.3 Reduction For the fina reduction from Panar 3Sat it remains to put together a gadgets according to the grid ayout of the given variabe cause graph G ϕ. We pace a variabe gadgets on the same y-coordinate, and a cause gadgets and pipes ie beow and above the variabes JoCG 7(1), ,

17 cause turn chain eft pipe cause midde pipe right pipe variabes variabes (a) (b) Figure 13: A ayout of a part of a variabe cause graph (a) and a sketch of the corresponding gadget pacement for the reduction (b). and form three-egged combs as in the input ayout of G ϕ. The overa structure of the gadget arrangement is sketched in Figure 13. At first gance it seems as if we might have a probem in describing the anchor coordinates with poynomiay bounded precision, since, for exampe, each variabe gadget requires us to pace anchors with distance 2. By encoding the anchor coordinates with agebraic numbers, however, we can avoid this probem. Reca that a our constructed gadgets are either aigned with the coordinate axes directy or they are soped by 15, 30, 45, or 60 with respect to an axis-parae ine through a reference point. Now it is we known that the sine and cosine vaues of these anges can a be expressed as agebraic numbers in the number fied Q( 2, 3). Hence, if we choose, say, the eftmost anchor of the eftmost variabe gadget as the origin, we can express the coordinates of any other anchor with respect to this origin as an agebraic number in Q( 2, 3). Theorem 1. MaxTota is NP-hard in the 1R hard-confict mode even if a abes are unit squares and their anchor points are their ower-eft corners. Proof. For a given panar 3-SAT formua ϕ we construct the MaxTota instance as described above. By construction of the hard conficts in our basic buiding bocks (chains and turns), we know that in any optima abeing each abe in such a buiding bock has exacty two choices for its active range, both of equa ength. Simiary, the inner abes of the cause gadgets have three choices for their active range, a of equa ength. Hence we can sum these upper bounds on the engths of the active ranges for a abes in our entire construction to obtain an upper bound K on the tota activity. It remains to show that ϕ is satisfiabe if and ony if this upper bound K can be achieved as the tota activity of our construction. In fact, in any basic buiding bock the respective upper bound on the tota activity is achieved if and ony if a active ranges are assigned correcty, i.e., corresponding to a truth assignment. Hence by Lemma 8 every unsatisfied cause forces its inner abe to have an active range of ony π/2 instead of the upper bound 2π/3. Thus we know that ϕ is satisfiabe if and ony if the MaxTota instance has a tota activity of at east K (in fact, the tota activity must be exacty K since this is aso an upper bound). Constructing and pacing the gadgets can be done in poynomia time and space, which concudes the proof. JoCG 7(1), ,

18 We note that the same construction as for the NP-hardness of MaxTota can aso be appied to prove NP-hardness of MaxMin. The maximay achievabe minimum ength of an active range for a satisfiabe formua is 2π/3, whereas for an unsatisfiabe formua the maximay achievabe minimum ength is π/2 due to Lemma 8. This observation aso yieds that MaxMin cannot be efficienty approximated within a factor of 3/4, uness P = NP. We summarize this in the foowing coroary. Coroary 1. MaxMin is NP-hard in the 1R hard-confict mode even if a abes are unit squares and their anchor points are their ower-eft corners. Moreover, there exists no efficient approximation agorithm with an approximation factor arger than 3/4 for this probem, uness P = NP. In the conference version of this paper [7] we erroneousy caimed that MaxTota is contained in NP and thus the (corresponding decision) probem is NP-compete. Athough it seems that due to the discretization emma we can simpy guess and verify a soution in poynomia time, we encounter the probem that in order to find the vaue of the guessed soution we potentiay need to sum up irrationa numbers. This, unfortunatey, cannot be done in poynomia time on a Turing machine, and hence, we fai to show membership of MaxTota in NP. 5 Approximation Agorithms for MaxTota In the previous section we have estabished that MaxTota is NP-hard. Uness P = NP we cannot hope for an efficient exact agorithm to sove the probem. In the foowing we devise a 1/4-approximation agorithm for MaxTota and then refine it to an EPTAS. For both agorithms we initiay assume that abes are congruent unit-height rectanges with constant width w 1 and that the anchor points are the ower-eft corners of the abes. Afterwards we argue how to drop this restriction. Let d be the ength of the abe s diagona, i.e., d = w Before we describe the agorithms we state one important requirement for our agorithms and three derived properties that appy even to the more genera abeing mode, where anchor points are arbitrary points within the abe or on its boundary, and where the ratio of the smaest and argest width and height, as we as the aspect ratio are bounded by constants: (i) the number of anchor points contained in any rectange is at most proportiona to its area, (ii) each abe has conficts with O(1) other abes, (iii) each abe has O(1) confict events with other abes, and finay, (iv) there is an optima MaxTota soution where a active ranges are bounded by abe events. Property (i) is a fundamenta requirement for our agorithms and demands that the point set P is not too dense. In fact, property (i) is usuay true in practice, e.g., if we JoCG 7(1), ,

19 assume that P is a set of features that admits a confict-free abeing in some static view of the map M as we show in Lemma 9. Lemma 10 proves property (ii) using a simpe packing argument. Property (iii) foows from Property (ii) and Lemma 1. Property (iv) foows immediatey from Lemma 4. Lemma 9. If M is an input map in which the abe set L consists of pairwise disjoint abes (corresponding to a vaid abeing of the entire point set P at rotation ange 0) then the number of anchor points in the interior or on the boundary of any rectange R with width W and height H is at most proportiona to the area of R. Proof. We consider the non-rotated map M, in which, by assumption, a abes are pairwise disjoint. Let the smaest abe height be h min, the smaest abe width be w min and the smaest abe area be a min. There can be at most 2W/w min + 2H/h min independent abes intersecting the boundary of R such that their anchor points are contained in R. A remaining abes with an anchor point in R must be competey contained in R, i.e., there can be at most W H/a min such abes. Hence, the number of anchor points in R is bounded by a constant. Lemma 10. Each abe has conficts with at most a constant number of other abes. Proof. For two abes and to have a confict their outer circes need to intersect and thus the maximum possibe distance between their anchor points is bounded by twice the maximum diameter of a abes in L. By the assumption that the height ratio, width ratio, and aspect ratio of a abes in L is bounded by a constant this diameter is bounded by a constant, too. Hence we can define for each abe a constant-size area around its anchor point containing a reevant anchor points. By Lemma 9 this area contains ony a constant number of anchor points. 5.1 A 1/4-approximation for MaxTota The basis for our agorithm is the ine stabbing or shifting technique by Hochbaum and Maass [11], which has been appied before to static abeing probems for (non-rotating) unitheight abes [1, 21]. Consider a grid G where each grid ce is a square with side ength 2d. We can address every grid ce by its row and coumn index. Now we can partition G into four subsets by deeting every other row and every other coumn with either even or odd parity. Within each of these subsets we have the property that any two grid ces have a distance of at east 2d. Thus no two abes whose anchor points ie in different ces of the same subset can have a confict. For an iustration see Figure 14. For a ce c we ca the area that is not contained in c but within distance of at most d to c the border of c. To compute an optima soution for the abes contained in c we must aso consider anchor points that ie in this area since during a fu rotation a abe may intersect such an anchor point. We have marked the border of the ce in row 2 and coumn 3 in bue in Figure 14. We say that a grid ce c covers a abe if its anchor point ies inside c. By Property (i) ony O(1) abes are covered by a singe grid ce. Combining this with Properties (ii) JoCG 7(1), ,