ON CHARACTERIZING TERRAIN VISIBILITY GRAPHS

Transcription

1 ON CHARACTERIZING TERRAIN VISIBILITY GRAPHS William Evans, and Noushin Saeedi Astrat. A terrain is an x-monotone polygonal line in the xy-plane. Two verties of a terrain are mutually visile if and only if there is no terrain vertex on or aove the open line segment onneting them. A graph whose verties represent terrain verties and whose edges represent mutually visile pairs of terrain verties is alled a terrain visiility graph. We would like to find properties that are oth neessary and suffiient for a graph to e a terrain visiility graph; that is, we would like to haraterize terrain visiility graphs. Aello et al. [Disrete and Computational Geometry, 14(3): , 1995] showed that all terrain visiility graphs are persistent. They showed that the visiility information of a terrain point set implies some ordering requirements on the slopes of the lines onneting pairs of points in any realization, and as a step towards showing suffiieny, they proved that for any persistent graph M there is a total order on the slopes of the (pseudo) lines in a generalized onfiguration of points whose visiility graph is M. We give a muh simpler proof of this result y estalishing an orientation to every triple of verties, refleting some slope ordering requirements that are onsistent with M eing the visiility graph, and prove that these requirements form a partial order. We give a faster algorithm to onstrut a total order on the slopes. Our approah attempts to larify the impliations of the graph theoreti properties on the ordering of the slopes, and may e interpreted as defining properties on an underlying oriented matroid that we show is a restrited type of 3-signotope. 1 Introdution Prolems related to geometri visiility have arisen from appliations in graphis and motion planning in rootis. In graphis, for example, the hidden line prolem for omputer-drawn polyhedra is to determine whih edges, or parts of edges, of a polyhedron are visile from a given vantage point. In rootis or more speifially motion planning, we would like to find the shortest path etween two positions for a root without hitting the ojets around. Visiility prolems inlude the well-known art-gallery prolems: planning a path for a guard so that it an see the entire art gallery, or determining the minimum numer of the guards whih together an see the whole art gallery. A deeper understanding of the ominatorial struture of visiility etween geometri ojets may help to address prolems involving visiility in omputational geometry. The results of the paper appeared in the seond author s Master s thesis [46]. Researh supported y NSERC Disovery grant. Department of Computer Siene, University of British Columia, {will,noushins}@s.u.a JoCG 6(1), ,

2 A visiility graph is a fundamental ominatorial struture whih has proven useful for addressing suh prolems. The verties of a visiility graph orrespond to geometri omponents suh as points or line segments. There is an edge etween two verties of the graph if the omponents are visile to eah other. There are variations in the definition of visiility depending on the underlying appliation. For the lass of art gallery prolems, the vertex visiility graph of polygons is more ommonly used. Here, the geometri omponents are the verties of the polygon, and two verties are visile if the line segment onneting them is in the interior or along the oundary of the polygon. Throughout this paper, y the term visiility graph, we mean the vertex visiility graph of the geometri ojet unless otherwise stated. Ideally, we would like to fully understand the ominatorial properties of visiility in polygons. We would like to find properties that are oth neessary and suffiient for a graph to e a visiility graph; that is, we would like to haraterize, graph theoretially, visiility graphs. If a graph satisfies the suffiient graph theoreti properties, it is realizale as a visiility graph of a polygon. Determining whether a graph with ertain properties is the visiility graph of a simple polygon is known as visiility graph reognition or realizaility. The atual drawing of the point set whose visiility graph is the desired graph is alled reonstrution. Studying the reognition and haraterization of visiility graphs may help us find more effiient algorithms for prolems related to geometri visiility. Although there are some partial results for restrited polygons, no haraterization of visiility graphs of simple polygons is known. The troule appears to e proving the suffiieny of ertain properties, for whih a two step approah might work: First, prove ominatorial onditions on point sets that an represent graphs with these properties; and seond, show that suh point sets an e geometrially realized to form the required visiility graph. The first step yields a haraterization of visiility graphs ut only in a generalized, non-geometri setting. Here the ominatorial onditions may e interpreted in the language of oriented matroids. The seond step is then to realize a lass of oriented matroids. Not all oriented matroids are realizale (the realizaility prolem is NP-hard), however, it may e possile to show that all matroids in this lass are realizale. 1.1 Prolem statement A one-dimensional terrain is an x-monotone polygonal line in the xy-plane. The endpoints of the terrain line segments mark terrain verties. Two verties of a terrain are mutually visile if and only if there is no terrain vertex on or aove the open line segment onneting them. The visiility graph of a terrain with n verties is the undireted graph with vertex set {1, 2,..., n} and edges {{a, } terrain verties a and are mutually visile}. We assume that the terrain visiility graph is ordered meaning that the ordering of the verties along the terrain agrees with the vertex numering. It is relatively simple to show that all terrain visiility graphs satisfy three properties desried in Setion 3. It remains hallenging to show that all graphs with these three properties, whih are alled persistent graphs (see Definition 3), are terrain visiility graphs. In fat, though Aello et al. [4] laimed this result, a omplete proof has never een pulished. JoCG 6(1), ,

3 Aello et al. studied ore indued sugraphs of the visiility graphs of stairase polygons (definitions at the start of Setion 2.2), whih are equivalent to ordered terrain visiility graphs. They showed that the visiility information of a given terrain vertex implies some ordering requirements on the slopes of the lines onneting this vertex to the others in any realization. A maximal hain in the weak Bruhat order (or equivalently, a alaned taleau, whih is defined in Setion 4.1) may e used to express these slope ordering requirements. As a step towards showing suffiieny, they provided a Θ(n 5 )-time algorithm that finds one maximal hain (of perhaps many) in the weak Bruhat order onsistent with a given persistent graph. Their algorithm starts with a representative maximal hain of a lique graph and repeatedly performs ompliated operations on the hain, in order to generate a maximal hain inrementally loser to that of the desired persistent graph. The idea was to use this maximal hain to onstrut a point set realizing it. However, the additional step turned out to e triky. 1.2 Generalized Conguration A alaned taleau may e viewed as representing a total order on the slopes of pseudolines that onnet pairs of points in a generalized onfiguration (definitions follow). A pseudoline is a simple urve that separates the plane. An arrangement of pseudolines is a olletion of pseudolines, suh that eah pair of them meet in exatly one point, where they ross. Let P e a set of points in the Eulidean plane, and let L e an arrangement of pseudolines suh that every pair of points in P lie on exatly one pseudoline, and eah pseudoline in L ontains exatly two points of P. Then the pair (P, L) is a generalized onfiguration of points in general position (in general position indiates that no three points of P lie on the same pseudoline of L). The onept of the slope order may also e generalized to pseudolines. If a pseudoline arrangement is interseted with a vertial line suh that all intersetion points of the arrangement lie to its right, then the order in whih the pseudolines ross the vertial line (dereasing y the y-oordinates of the rossings) is the (inreasing) slope order of the pseudolines [52]. 1.3 Desription of results We give a streamlined proof of Aello et al. s result [4] (that there is a representative alaned taleau for every persistent graph). Our approah is to translate the onstraints on slope ordering imposed y the persistent graph into orientations on triples of verties and to ome up with an orientation on all vertex triples that agrees with these onstraints (Setion 4.3.1). We interpret our orientation as defining properties on an underlying oriented matroid, and show that our orientation is a restrited type of a 3-signotope. This, together with the work of Felsner and Weil [28] on ayliity of signotopes, gives an immediate proof of the result (Setion 4.3.3). In fat, this is a slightly stronger result than that of Aello et al. s. We also give an alternate self-ontained proof (Setion 4.3.4) that larifies the impliations of the graph theoreti properties on the slope orders in more detail. This has the JoCG 6(1), ,

4 potential to determine additional onstraints on the slopes that may aid in realization. Felsner and Weil give an astrat ominatorial proof that the graphs assoiated with signotopes are ayli in general. Our diret proof may e viewed as an alternative and perhaps more intuitive proof for the ayliity of suh graphs when restrited to 3- signotopes. While the sustantial ideas of oth proofs are the same, our proof explains in more detail the manner in whih ertain slope orderings entail others. Our proofs imply a Θ(n 3 )-time algorithm to onstrut a representative alaned taleau (Setion 4.3.5), whih improves on the Θ(n 5 ) algorithm of Aello et al. [4]. Moreover, the slope orders imposed y our orientation an e preserved when extending the graph y additional verties. Our orientation also forids ertain sustrutures in the point set realizing it. These properties may help prove realizaility (Setion 4.3.6). Lastly, we put our work and the previous results in the ontext of oriented matroids, and give a thorough omparison of these studies (Setion 5). 2 Related work Visiility graphs have een extensively studied. Visiility may e defined among the verties of a polygon, line segments in the plane, or various other geometri ojets in two or higher dimensions. O Rourke [41], and Ghosh and Goswami [33] give an exellent review of researh on visiility graphs. Ghosh s ook [32] is also a good soure of information on visiility graphs. Charaterizing visiility graphs of simple polygons and finding algorithms to reognize them seem hallenging. There is no polynomial-time algorithm known to reognize visiility graphs. Nor is the prolem known to e NP-hard, or even in NP. Everett [25] shows that the prolem an e solved in PSPACE. The lass of visiility graphs does not lie in any of the well-known lasses of graphs suh as planar graphs, hordal, irle or perfet graphs [25, 31]. Everett [25] shows that there is no finite set of foridden indued sugraphs that haraterizes visiility graphs. Results on visiility graphs so far have involved restriting the lass of graphs, or restriting the lass of polygons, or adding extra information to the graph. 2.1 Restriting the lass of graphs ElGindy [24] shows that any maximal outerplanar graph is a visiility graph, and provides a linear time emedding algorithm to onstrut a uni-monotone polygon whose visiility graph is the desired maximal outerplanar graph. A monotone polygonal hain is a set of verties in the plane, ordered along some diretion, with onseutive verties joined. A polygon is monotone if it an e roken into two monotone polygonal hains (oth ordered along the same diretion). A uni-monotone polygon is a monotone polygon where one hain onsists of a single edge. A terrain, whose visiility properties we study here, is the same as a monotone polygonal hain (monotone in the horizontal diretion). Colley [20] extends the range of graphs that an e identified as visiility graphs. He JoCG 6(1), ,

5 defines a new lass of graphs, tree of liques graphs, and extends ElGindy s algorithm to reognize these graphs y emedding a uni-monotone polygon with the required visiility. Both ElGindy s and Colley s algorithms an e modified to reate a uniform uni-monotone polygon (that is, a uni-monotone polygon whose verties are uniformly spaed in the diretion of monotoniity). Not every uniform uni-monotone polygon has a tree of liques visiility graph. 2.2 Restriting the lass of polygons A stairase polygon is a polygon onsisting of alternate horizontal and vertial edges forming a polygonal hain leading to the right and down, and a path of one horizontal and one vertial edge onneting its endpoints. These polygons are also known as orthogonal onvex fans. A onvex fan is a polygon that has a onvex vertex, alled the kernel, visile to all other verties. An orthogonal onvex fan is a onvex fan onsisting of only horizontal and vertial edges. The ore of a stairase polygon is the set of reflex verties plus the verties adjaent to the kernel vertex. A uniform step-length stairase polygon is a stairase polygon whose verties are evenly spaed with respet to the horizontal diretion. Aello and Egeioglu [2] give a polynomial time reognition algorithm, using linear programming, to reognize visiility graphs of uniform step-length stairase polygons. Using this method, they show that there exist graphs that are visiility graphs of stairase polygons ut not realizale as uniform step-length stairase polygons. Colley [20, 21] proves a strong relationship etween the visiility graphs of unimonotone polygons and stairase polygons. He uses the result to show that reognizing the visiility graph of a stairase polygon is equivalent, under linear-time redution, to the prolem of reognizing the visiility graph of a monotone polygonal hain, where the additional information of the order of the verties on the hain is given. The vertex ordering of the ore of a stairase polygon is determined y the visiility graph of the stairase polygon. A ruial part of his argument is that the ore indued sugraph of the visiility graph of a stairase polygon is idential to the visiility graph of a monotone polygonal hain (or equivalently a terrain visiility graph), and vie versa. Using this and the result from Aello and Egeioglu [2], Colley shows that the visiility graphs of uniform uni-monotone polygons are a strit suset of the visiility graphs of general uni-monotone polygons, if the outside fae of the polygon is fixed. There have een few results that give a graph theoreti haraterization of visiility graphs for restrited lasses of polygons. Everett and Corneil [26] haraterize the visiility graphs of 1-spiral polygons and give a linear time algorithm for reognizing them. Colley et al. [22] haraterize the visiility graphs of towers and present a linear time algorithm to reognize them. Choi et al. [19] also haraterize these graphs, where they use the term funnel instead of tower. They give a linear time algorithm to reonstrut the funnel from its visiility graph. JoCG 6(1), ,

6 2.3 Adding extra information Many results on visiility graphs involve adding extra information to the graphs. Ghosh [30] onjetures three neessary onditions for reognizing visiility graphs of simple polygons, when the Hamiltonian yle forming the polygon oundary is speified. Everett [25] gives a ounterexample to this onjeture and suggests a stronger version of the third neessary ondition, whih Srinivasaraghavan and Mukhopadhyay [48] prove is indeed neessary. Aello et al. [9] show that, even with the stronger version of the third ondition, the neessary onditions are insuffiient. Ghosh [31] identifies another neessary ondition to irumvent the new ounterexample and onjetures that the four neessary onditions are suffiient, ut Streinu [51] later proves Ghosh s onjeture is false. Coullard and Luiw [23] introdue further neessary onditions for a graph to e a visiility graph. They develop a new strutural property of visiility graphs: Eah 3- onneted omponent of a visiility graph has a vertex ordering in whih every vertex is adjaent to a previous 3-lique; that is, eah 3-onneted omponent of a visiility graph has a 3-lique ordering. The weaker result that eah vertex is adjaent to a previous 2-lique is a onsequene of polygon triangulation. The 3-lique ordering property is not suffiient. The property an e tested in polynomial time, and is used to give an algorithm for the distane visiility graph prolem, whih is the prolem of whether an edge-weighted graph is the visiility graph of a simple polygon with the weights as Eulidean distanes. ElGindy [24] onjetures a haraterization of visiility graphs of onvex fans. He suggests a deomposition strategy and gives an algorithm to hek whether a graph is the visiility graph of a onvex fan, when the Hamiltonian yle forming the oundary is known. However, the reonstrution appears triky and the orretness of the algorithm is not lear. Aello et al. [3, 4] laim to have haraterized visiility graphs of stairase polygons (or equivalently orthogonal onvex fans) in a series of two papers, only one of whih has appeared in the literature. The preliminary results y Aello et al.[8] are preursors to these two papers. The diffiult part of suh a haraterization is to haraterize the ore indued sugraphs of these visiility graphs. The non-ore verties are easy to determine from the graph. They identify the Hamiltonian yle and as a result the ordering of the ore verties in the graph. Aello et al.[4] and Aello [1] present a neessary property for suh ore indued sugraphs, whih they all persistent property. They show that the visiility etween ore verties (in a stairase polygon) implies some ordering requirements on the slopes of the lines that onnet pairs of these verties in any realization. They approah the prolem of whether the persistent property is suffiient, given the ordering of the ore verties, y onstruting a total order on the slopes of (pseudo) lines that onnet pairs of ore verties in a generalized onfiguration suh that the slope order is onsistent with the desired visiility graph. The non-ore verties would e easy to add to any ore realization. Aello et al. [4, 8] laim that the proposed slope order is realizale y a point set ut a omplete proof has not een pulished. We desrie their work in more detail in Setion 4.2. Following Ghosh s result [30], Aello and Kumar [5, 7] study visiility graphs of simple polygons, when the Hamiltonian yle forming the oundary of the polygon is given. JoCG 6(1), ,

7 They define a new lass of graphs alled quasi-persistent graphs, whih they show is equivalent to the lass of graphs satisfying the first two neessary onditions of Ghosh [30]. Using a geometri interpretation of a polygon realizing a quasi-persistent graph, they determine verties that lok the line of sight etween pairs that are not mutually visile. This gives a loking vertex assignment. (Different polygons with the same visiility graph may have different loking vertex assignments.) They show that a loking vertex assignment, if determined y a polygon realization, satisfies four neessary onditions. The last three onditions are ased on properties of Eulidean shortest paths (as the shortest path etween a pair that is not mutually visile is determined y the loking vertex assignment of the pairs on the path). Ghosh [31] shows all four neessary onditions of Aello and Kumar follow from his third and fourth onditions. To address the realizaility prolem, Aello and Kumar introdue an oriented matroid approah: first find the ominatorial properties on the point set orresponding to the verties of the visiility graph (that is represented y an oriented matroid); then deide whether suh a point set is realizale. In partiular, they show how to onstrut a uniform rank 3 oriented matroid for every quasi-persistent graph satisfying the four onditions, whih if affinely realizale yields a simple polygon with the desired visiility graph. Aello and Kumar [6] show that these onditions are suffiient to reonstrut a polygon from the graph when restrited to 2-spiral polygons. O Rourke and Streinu [43] introdue a new polygon visiility graph, the vertexedge visiility graph, that represents visiility etween verties and edges. They suggest that the additional geometri information suh graphs provide may simplify the prolem of haraterizing them. They show that a vertex-edge visiility graph determines the onvexity of verties, the vertex visiility graph, and, for eah vertex, the partial loal sequene (definition elow) and the shortest path tree. The loal sequene for a vertex v is the irular sequene of all other verties as they are enountered y a rotating line through v. The partial loal sequene ontains only the visile verties. The olletion of all loal sequenes is a version of what Goodman and Pollak [34] alled a luster of stars [49], whih forms an affine (or ayli) uniform rank 3 oriented matroid, whose topologial representation is a generalized onfiguration of points [34]. The set of all loal sequenes of a generalized onfiguration of points determines the hirotope information, or equivalently the set of all triple orientations, of the point set. The orientation of a triple (i, j, k) shows whether k is to the right or to the left of the pseudoline through i to j. The same is true when pseudolines are straight-lines. The set of all triple orientations determines the order type. O Rourke and Streinu [42] generalize the notion of straight-line visiility to visiility along pseudolines, whih they all pseudo-visiility. The idea of pseudo-visiility omes from the onept of duality etween pseudoline arrangements and generalized onfigurations of points. They give a omplete haraterization of vertex-edge (pseudo) visiility graphs of pseudo-polygons. They define a prediate on any triple of verties (the prediates math the hirotope definition of Aello and Kumar [7]). They show the prediates satisfy Knuth s CC system axioms [38]. CC systems are equivalent to uniform rank 3 ayli oriented matroids [38], whih in turn are equivalent to Goodman and Pollak s generalized onfig- JoCG 6(1), ,

8 urations of points in general position [12]. As a onsequene of the relationship etween vertex-edge visiility graphs and vertex visiility graphs, they show that the reognition prolem for vertex visiility graphs of pseudo-polygons is in NP (the same prolem with straight-line visiility is only known to e in PSPACE). Streinu [49] gives a haraterization of lusters of stars of generalized onfigurations of points, and provides effiient algorithms for reognizing them. Her haraterization onditions indiate that an orientation (lokwise or ounterlokwise) of every triple that is onsistent over a set of loal sequenes (that is, oeys a generalized transitivity law) is realizale as a generalized onfiguration of points, for some ordering of the point set. Knuth s CC systems [38] an also e interpreted as haraterizing the loal sequenes of generalized onfigurations of points. The onept of pseudo-visiility detahes the strethaility question from the ominatorial aspets of the prolem. A pseudoline arrangement (or equivalently an ayli uniform rank 3 oriented matroid) is strethale or realizale if it is isomorphi to a straightline arrangement. The main diffiulty in fully haraterizing vertex-edge visiility graphs of (straight-line) polygons is to deide whether a ertain lass of ayli uniform rank 3 oriented matroids is strethale. It is well-known that strethaility of pseudoline arrangements, in general, is NP-hard [39, 47]. However, there exist various tehniques to prove strethaility for partiular instanes [16, 17, 13, 44, 14, 15, 45]. O Rourke and Streinu [42] remark that the lass of ayli uniform rank 3 oriented matroids generated y the vertex-edge pseudovisiility graphs is a strit sulass of all ayli uniform rank 3 oriented matroids, so it may e possile to haraterize or reognize them effiiently. Streinu [50] shows that the lass of vertex-edge (straight-line) visiility graphs is properly ontained in the lass of vertex-edge pseudo-visiility graphs. She introdues star-like pseudo-polygons and shows that a star-like vertex-edge pseudo-visiility graph may not e realizale. However, sine vertex-edge pseudo-visiility graphs ontain more geometri information than pseudo-visiility graphs, it is possile that a pseudo-visiility graph is assoiated with several (possily exponentially many) vertex-edge pseudo-visiility graphs, some of whih are strethale and some not [51]. For instane, all star-like pseudovisiility graphs are realizale [50]. Later, Streinu [51] shows an infinite family of pseudovisiility graphs that are not realizale. Sine the graphs in this family satisfy the neessary onditions of O Rourke and Streinu [42], Aello and Kumar [7], and Ghosh [31], this implies that these neessary onditions are not suffiient to haraterize (straight-line) visiility graphs. Everett et al. [27] study other ominatorial ojets (ontaining more information than visiility graphs) that desrie the struture of simple polygons, whih they all the staing information. The staing information stores ertain information aout the intersetions that eah line onneting two verties makes with the other edges of the polygon. O Rourke [40], and Jakson and Wismath [36] study weaker variants of suh ojets for orthogonal polygons that involve the horizontal and vertial visiility information. Jakson and Wismath [36] show how to reonstrut an orthogonal polygon from oth the internal and external horizontal and vertial visiility information. JoCG 6(1), ,

9 3 Neessary properties for terrain visiility graphs We first define the graph properties that we use, and then present the set of onditions satisfied y all terrain visiility graphs. For integers a and we use the notation [a..] to indiate the interval of all integers etween and inluding them. The two numers a and are alled the endpoints of the interval. To exlude an endpoint from the interval we replae the assoiated raket with a parenthesis. For example (i..j] is {x N i < x j}. We use [n] for [1..n] for suintness. Definition 1. An ordered graph G = ([n], E) has the X-property if for every four verties a < < < d, if {a, } E and {, d} E then {a, d} E. This property is alled inversion omplete y Aello [1]. Definition 2. An ordered graph G = ([n], E) has the ar-property if for every edge {a, } E where a + 1 <, there exists a vertex (a..) suh that {a, } E and {, } E. In an ordered graph with Hamiltonian path 1, 2,..., n, this property is an ordered version of hordality [1, 31]: every ordered yle of length at least four has a hord. Definition 3. An ordered graph G = ([n], E) that has oth the X-property and the arproperty and ontains the Hamiltonian path 1, 2,..., n is alled persistent. Aello et al. [4] initially defined persistent graphs in 1995 in a different and slightly inorret manner so that the X-property was not guaranteed (inorret in the sense that some properties that they assume from their definition are not true). Aello modified the definition in 2004 [1] to inlude this property (whih he alled inversion ompleteness). His susequent definition gives the same lass of graphs as our definition does. We prove that ordered terrain visiility graphs are persistent. Colley [20] proves that the ore indued sugraph of the visiility graph of a stairase polygon is idential to a terrain visiility graph, and vie versa. Aello et al. [4] show that the ore indued sugraph of the visiility graph of a stairase polygon (and as a result a terrain visiility graph), with respet to its ordering, is persistent. Here, for ompleteness of results, we reprove the neessary onditions for terrain visiility graphs y a simpler and more geometri approah. Our proof relies on two lemmas. The first is alled the Order Claim in the literature [11, 18, 37], and the seond is alled the Midpoint Claim y King [37]. Lemma 1. Ordered terrain visiility graphs have the X-property. Proof. Consider four verties a < < < d in an ordered terrain visiility graph G = ([n], E), suh that {a, } E and {, d} E. Sine {a, } E, we know the terrain does not interset the half-strip aove the segment a. Denote the half-strip aove the segment uv y H + (uv). Similarly, we know the terrain does not interset H + (d). Hene is elow the segment a and is elow the segment d, whih means the two segments a and d interset. Thus the line segment ad lies in the region H + (a) H + (d). Therefore, the terrain does not interset H + (ad), whih implies {a, d} E. See Figure 1. JoCG 6(1), ,

10 d a Figure 1: X-property in ordered terrain visiility graphs. Lemma 2. Ordered terrain visiility graphs have the ar-property. Proof. Consider an ordered terrain visiility graph G = ([n], E). For any edge {a, } E where a + 1 <, the terrain indued on the verties in [a..], together with the edge {a, }, reates a simple polygon. The fat that every simple polygon admits a triangulation implies the existene of a vertex (a..) suh that {a, } E and {, } E. See Figure 2. a Figure 2: Bar-property in ordered terrain visiility graphs. We know that ordered terrain visiility graphs ontain the Hamiltonian path 1, 2,..., n. Moreover, y Lemma 1 and Lemma 2, they also satisfy oth the X-property and the arproperty. Thus we onlude the following theorem (whih is equivalent to Theorem 3.5 y Aello et al. [4]). Theorem 1. Ordered terrain visiility graphs are persistent. 4 On the suieny of the persistent property Ideally, we would like to show that the persistent property is suffiient to imply that the graph is a terrain visiility graph, and to e ale to reover a terrain from a given persistent graph; ut this seems hallenging. Aello et al. [4] showed that the visiility information of a terrain point set implies some ordering requirements on the slopes of the lines onneting pairs of points in any realization, and onstruted a total order on the slopes of the lines in a generalized onfiguration of points with the desired visiility. It is unknown whether JoCG 6(1), ,

11 there is a point set that realizes the resulting slope order. It is worth mentioning that a slope ordering onsistent with the desired terrain visiility is not unique. We give a muh simpler proof that the slope ordering requirements otained from any persistent visiility graph form a partial order. Our approah is to estalish an orientation on every triple of verties, refleting some slope ordering requirements, suh that it is onsistent with the desired visiility information. Our proof also gives a faster algorithm for onstruting a total order on the slopes. Here, we first introdue the terminology that we use for the representation of the slope ordering. We next desrie the overall approah of Aello et al. [4] riefly. We then show that our orientation is a restrited type of a 3-signotope (Setion 4.3.3), whih together with the work of Felsner and Weil [28], gives an immediate proof of the result. We also give an alternate self-ontained proof (Setion 4.3.4) that larifies the impliations of the graph theoreti properties on the slope orders, whih may help in approahing realizaility. Lastly, we present a Θ(n 3 )-time algorithm for onstruting a total order on the slopes (Setion 4.3.5) and onlude y disussing properties of our orientation. 4.1 Representation of the slope order We use terminology similar to that used y Aello et al. [4] for the representation of the slope order of the (pseudo) lines onneting pairs of points in a (generalized) onfiguration. A taleau of size n is a two-dimensional array of n 1 rows (indexed from 2 to n) where row r ontains r 1 entries (indexed from 1 to r 1), and whose entries are the integers 1, 2,..., ( n 2). For a taleau T, we refer to the entry in row r and olumn as T [r, ]. Note that r >. Consider a non-degenerate point set {1, 2,..., n}. We may represent the slope ordering of the lines onneting all pairs of the points y a taleau T of size n, suh that T [j, i] is the rank of the slope of the line through i and j. (In other words, T [j, i] = s if and only if the slope of the line through i and j is the s-th smallest of all suh slopes.) We know that every three points a < < have either a positive orientation (that is, lies elow the segment a) or a negative orientation (that is, lies aove the segment a). This implies that, in the taleau T representing the slope ordering, we have either T [, a] < T [, a] < T [, ] or T [, a] > T [, a] > T [, ]. (This estalishes Lemma 3.1 of Aello et al. [4].) For three integers a < <, we say the triple T [, a], T [, a], and T [, ] is oriented positively if T [, a] < T [, a] < T [, ]. Similarly, the triple is oriented negatively if T [, a] > T [, a] > T [, ]. The triple is alaned if either T [, a] < T [, a] < T [, ] or T [, a] > T [, a] > T [, ]. A alaned taleau is a taleau whose triples are all alaned. The skeleton S T of a taleau T is a two-dimensional array of the same dimensions as T where { 1 if = a + 1 or T [, a] > T [, a] for all (a..), S T [, a] = 0 otherwise. Figure 3 shows a alaned taleau and its skeleton. JoCG 6(1), ,

12 Figure 3: A alaned taleau and its skeleton. An n-triangle is the strit lower triangle of an n n (0, 1)-matrix. For an n-triangle M, M[r, ] is the entry in row r and olumn, and M[[..d]; [a..]] is the matrix formed y the rows [..d] and the olumns [a..]. A persistent n-triangle is the n-triangle of the adjaeny matrix of a persistent graph. A skeleton S T of a alaned taleau T of size n an e interpreted as the n- triangle of an adjaeny matrix for an undireted graph with verties 1, 2,..., n and edges {{a, } S T [, a] = 1}. This graph is the skeleton graph of the alaned taleau. It is easy to show that the skeleton graph of a alaned taleau representing the slope order of a terrain point set is idential to the terrain visiility graph. We know S T [, a] = 1 if and only if = a + 1 or the slope of line a is greater than the slopes of all lines a, where (a..). This means that S T [, a] = 1 if and only if the terrain verties a and are adjaent in the terrain, or all terrain verties in (a..) are elow the line a; that is, terrain verties a and are mutually visile. (This argument is Lemma 3.2 in Aello et al. [4].) A taleau representing the slope order of a simple onfiguration of points is alaned (See Lemma 3.1 of Aello et al. [4]). This and Theorem 1 imply that the skeleton graph of a alaned taleau representing a terrain is persistent. Ideally, we would like to show that persistent graphs are visiility graphs of terrains. This is equivalent to showing that every persistent graph is the skeleton of a alaned taleau whose entries represent the ranks of the slopes etween pairs of points in a terrain. We have not proved this ut here we show that every persistent graph is the skeleton graph of a alaned taleau (Lemma 5.12 of Aello et al. [4]). Theorem 2. If M is a persistent n-triangle, then there exists a alaned taleau whose skeleton is idential to M. We will prove this theorem in Setion Aello et al.'s approah Aello et al. [4] prove that a graph is persistent if and only if it is the skeleton graph of a alaned taleau. They argue that a skeleton graph of a alaned taleau is persistent diretly y using the definitions. Here we only sketh their approah for the reverse diretion (that is, every persistent graph is the skeleton graph of a alaned taleau). JoCG 6(1), ,

13 For a persistent graph G = (V, E), they define an edge e to e reversile if the graph G = (V, E \ e) remains persistent. They use the idea of reversile edges to partially order persistent graphs so that they an generate any of them in a anonial manner. Given a persistent graph with n verties, they give an algorithm that starts from a lique of n verties and suessively removes a reversile edge until the desired persistent graph is generated. They use the asis of this algorithm to reonstrut a alaned taleau from a given persistent graph. Namely, they start from a alaned taleau whose skeleton represents a lique, and perform operations on the taleau entries so that the underlying skeleton eomes inrementally loser to the desired graph. They define flush and augmentation operations, whih are ompositions of Coxeter type I and type II transformations, on a alaned taleau. Eah flipping of a 1 entry in the skeleton (that is, removing a reversile edge) is done through a sequene of flush and augmentation operations on the taleau and is ompliated. They estalish a loop invariant to show the orretness of their proposed algorithm. However, the loop invariant is not intuitive and requires a omplex proof. The overall omplexity of their algorithm is Θ(n 5 ). 4.3 Our main result We reprove that every persistent graph is the skeleton graph of a alaned taleau ut in a muh simpler way. By the definition of a taleau s skeleton, we know that the skeleton entries (whether 0 or 1) imply ertain inequality relations amongst the taleau entries. The following lemma summarizes these relations. Lemma 3. For a taleau T and an n-triangle M, S T = M if and only if 1. If M[, a] = 1, then = a + 1 or T [, a] > T [, a] for all (a..), and 2. If M[, a] = 0, then there exists (a..) suh that T [, a] > T [, a]. Proof. Lemma 3 follows diretly from the definition of the skeleton of a taleau. Using the ominatorial properties of a persistent graph G, we would like to derive a alaned taleau satisfying the onditions in Lemma 3, where M is the n-triangle of the adjaeny matrix of G. We show that the inequality relations required y Lemma 3 and y the alaned property give a partial order on the taleau entries; and as a result any taleau that realizes this partial order is alaned, with a skeleton graph idential to the given persistent graph. Our approah is to orient all taleau triples so that they are onsistent with Lemma 3. Sine our orientation is guaranteed to e alaned, to show the existene of suh a alaned taleau, we only need to show that our orientation forms a partial order on the taleau entries Orienting the triples We infer ominatorial properties on the struture of a persistent n-triangle, and use these strutural properties in determining the orientation of a taleau triple. JoCG 6(1), ,

14 Definition 4. Let M e an n-triangle. We define hook(a) to e the sustruture M[[..]; a] M[; [a..]], where a < <. The orner of hook(a) is M[, a]. The olumn-arm of hook(a) is M[[..]; a], and the row-arm of hook(a) is M[; [a..]]. The half-strit a-retangle is the sustruture M[(..]; [a..)], and is denoted y ret(a). ret(a) ret(xyz) means ret(a) is ontained in ret(xyz). (Figure 4 illustrates the half-strit a-retangle and hook(a).) ret(a) a Figure 4: The half-strit a-retangle (shaded) and hook(a) (old outline). M. The following lemma identifies the struture of hook(a) in a persistent n-triangle Lemma 4. If M is a persistent n-triangle then eah hook(a) is of one of the following forms (See Figure 5): The orner is 1, or Either the row-arm or the olumn-arm ontains only zeros, or M[, a] = M[, ] = 1 and all other entries in hook(a) are zeros. 1 a a a a Figure 5: Possile strutures of hook(a) in a persistent n-triangle. Proof. Suppose to the ontrary that there exists a hook(a) that does not have any of the aove properties. Thus, we have 1. M[, a] = 0; 2. there exists i (a..] suh that M[, i] = 1; and 3. there exists j [..) suh that M[j, a] = 1; JoCG 6(1), ,

15 suh that either i or j. But this ontradits M having the X-property on the four points a < i < j <, whih is impossile. It is easy to dedue the possile strutures of a half-strit a-retangle in a persistent n-triangle, from Lemma 4 and the X-property. Figure 6 illustrates these possile strutures with regards to hook(a). As shown in the piture, in the top four ases, eah half-strit a-retangle ontains a 1 entry whereas in the ottom four ases, eah half-strit aretangle ontains only zeros. 1 a a a a (a) ret(a) ontains a all a 1 0 all a 0 0 all a 0 0 all a () ret(a) ontains only zeros. Figure 6: Possile strutures of a half-strit a-retangle with regards to hook(a) in a persistent n-triangle. From the ar-property we onlude the following lemma. Lemma 5. Let M e a persistent n-triangle. For every a < < < d in [n], if oth ret(a) and ret(d) ontain only zeros then M[(..d]; [a..)] is a zero matrix. Proof. The k-upper triangle of a matrix is the olletion of the entries aove the k-th diagonal. We prove the ontrapositive. Suppose M[(..d]; [a..)] ontains a 1 entry. Let M[, a ] e a 1 entry lying on the k-th diagonal in the matrix M[(..d]; [a..)] suh that the k-upper triangle of the matrix ontains only zeros. It is easy to see that suh an entry exists. We have M[, a ] = 1 with a a < < <. Sine M is persistent, y the ar-property we know there exists (a.. ) suh that M[, a ] = 1 and M[, ] = 1. Sine the k-upper triangle in the matrix M[(..d]; [a..)] ontains only zeros, we infer. See Figure 7(a). If ret(d) ontains only 0 entries, then =. Hene M[, a ] = M[, a ] = 1, whih implies that there is a 1 entry in M[; [a..)]. See Figure 7(). Therefore either ret(a) or ret(d) ontains a 1 entry, whih onludes the proof. Let M e a persistent n-triangle. Given M, we define the orientation funtion α M from the 3-element susets of [n] to {+, } as follows: JoCG 6(1), ,

16 a a d 1 d aa d (a) The grey region ontains only zeros, and hene. aa d () ret(d) ontaining only zeros implies that =, and M[, a ] = 1. Figure 7: For a persistent M, if M[(..d]; [a..)] ontains a 1 then, y the ar-property, either ret(a) or ret(d) ontains a 1 as well. For every a < < in [n], let { + if ret(a) ontains a 1 entry, α M (a) = if ret(a) ontains only 0 entries. Oservation 1 (retangle ontainment). For ret(a) and ret(xyz) suh that ret(a) ret(xyz), we have 1.1. if α M (a) = +, then α M (xyz) = +, and 1.2. if α M (xyz) =, then α M (a) =. Oservation 2. Let M e a persistent n-triangle. For every a < < < d in [n], 2.1. if α M (a) = α M (d) =, then α M (ad) = α M (ad) = ; 2.2. if α M (a) = and α M (ad) = +, then α M (d) = +; and 2.3. if α M (d) = and α M (ad) = +, then α M (a) = +. Proof. Oservation 2.1 is a restatement of Lemma 5. Oservations 2.2 and 2.3 logially follow from Oservation 2.1. A taleau T agrees with orientation funtion α M if for all a < <, T [, a] < T [, a] < T [, ] if α M (a) = +, and T [, a] > T [, a] > T [, ] if α M (a) =. Figure 8 illustrates the inequality relations required in a taleau that agrees with α M. Suppose there exists a taleau T that agrees with α M. The following lemma shows that our orientation aptures oth the alaned property and the properties required to satisfy S T = M. JoCG 6(1), ,

17 a 1 + a 0 - Figure 8: The implied inequality relations etween the taleau entries (indiated y the old direted edges). Lemma 6. Let M e a persistent n-triangle. If there is a taleau T whose entries agree with orientation funtion α M, then 1. T is alaned, and 2. S T = M. Proof. We have eah taleau triple oriented either positively or negatively, and hene all triples are alaned. Therefore T is a alaned taleau. As illustrated in Figure 9, it is easy to oserve that having eah taleau triple oriented y α M implies that: 1. if M[, a] = 1, then = a + 1 or T [, a] > T [, a] for all (a..), and 2. if M[, a] = 0, then there exists (a..) suh that T [, a] > T [, a]. (A partiular is (a..) suh that M[, a] = 1 and M[x, a] = 0 for all x (..). Notie that suh always exists eause M[a + 1, a] = 1 due to the Hamiltonian path 1, 2,..., n, whih is guaranteed y the persistent property.) Thus the triple orientations indue the onditions in Lemma 3, whih implies S T = M. 1 a a a a + (a) ret(a) ontains a all a 1 0 all a 0 0 all a 0 0 all a - () ret(a) ontains only zeros. Figure 9: Possile triple orientations and their implied inequality relations over a taleau that agrees with the orientation are illustrated with regards to the assoiated half-strit retangles. JoCG 6(1), ,

18 If a taleau T agrees with α M then the entries of T have to oey a set of inequalities (as shown in Figure 9). From the aove lemma, we onlude that if the set of all these inequalities gives a partial order on the taleau entries, then any taleau realizing this partial order would e a alaned taleau with skeleton M. In the following, we prove that our orientation gives a partial order The M-taleau relation digraph D M We introdue a direted graph to represent the required inequality relations etween the taleau entries so that the taleau agrees with α M. We show that the resulting digraph is ayli if M is persistent, whih onludes the proof of Theorem 2. Definition 5. Let M e a persistent n-triangle. direted graph D M = (V M, E M ) suh that The M-taleau relation digraph is a V M = {r r, [n] and r > } (the vertex r represents the taleau entry in row r and olumn ), and E M = {(a, a) and (a, ) α M (a) = } {(, a) and (a, a) α M (a) = +} (the edges represent the inequality relations over the entries of a taleau that agrees with α M ; for a taleau T, the edge (a, a) represents that T [, a] > T [, a]). In the next setion, we show that orientation funtion α M is a signotope of a partiular type. This, oupled with a result from Felsner and Weil [28], proves the ayliity of D M Orientation funtion α M is a restrited type of signotope Felsner and Weil introdued signotopes in their study of sweeps of pseudoline arrangements. We first desrie their terminology. The notation ( [n]) k represents the k-element susets of [n]. Definition 6. Let +. For an integer r suh that 1 r n, an r-signotope on [n] is a funtion α : ( ) [n] r {, +} suh that for every (r + 1)-element suset P of [n] and all i, j, and k suh that 1 i < j < k r + 1 either α(p i ) α(p j ) α(p k ) or α(p i ) α(p j ) α(p k ), where P x denotes the set P minus the x-th largest element of P. We refer to this property as monotoniity. It is easy to restate Definition 6 for r = 3 as follows. Definition 7. A 3-signotope on [n] is a funtion α : ( ) [n] 3 {, +} suh that for every 4-element suset {a,,, d} of [n] with a < < < d, the orientation sequene (α(a), α(ad), α(ad), α(d)) is monotone (that is, it has at most one hange of sign). In other words, it is one of the eight olumns of the tale in Figure 10. We use this definition in proving the following Theorem. JoCG 6(1), ,

19 α(a) α(ad) α(ad) α(d) Figure 10: Monotone orientation sequenes. Theorem 3. Let M e a persistent n-triangle. The orientation funtion α M : ( ) [n] 3 {, +} where { + if ret(a) ontains a 1 entry α M (a) = otherwise, is a 3-signotope. Moreover, for every 4-tuple, the resulting orientation sequene exludes (,,, +) and (+,,, ). Proof. We need to show that for any 4-element suset P = {a,,, d} of [n] with a < < < d, the orientation sequene (α M (a), α M (ad), α M (ad), α M (d)) is monotone. Figure 11 illustrates the relationships etween ret(a), ret(ad), ret(ad), and ret(d). Let P x denote the set P minus the x-th largest element of P. The position i of the orientation sequene ontains α M (P i ). a d a d Figure 11: The relationships etween ret(a) (lightly shaded red), ret(ad) (antidiagonally striped red), ret(ad) (diagonally striped lue), and ret(d) (oldly shaded lue). We know either α M (d) = + or α M (d) =. Consider the following ases. Case 1. α M (d) = +: Sine ret(d) ret(ad), we onlude α M (ad) = +. Thus, we may get a non-monotone orientation sequene only if α M (ad) = and α M (a) = +. But this is impossile eause ret(a) ret(ad). Case 2. α M (d) = : Suppose we have a non-monotone orientation sequene. Thus the sequene ontains at least one + sign. We show that the rightmost + sign in the orientation sequene propagates all the way to the left, whih implies that the sequene is monotone. Note that, eause ret(a) ret(ad), if α M (a) = + then α M (ad) = +. Therefore, the rightmost + sign is either at position 2 or at position 3. We onsider eah ase elow. JoCG 6(1), ,

20 Case 2.1. α M (ad) = + and α M (ad) = : We know ret(ad) ontains a 1 entry whereas ret(ad) ontains only zeros. This implies that ret(a) ontains a 1 entry. Thus α M (a) = +, whih onludes this ase. Case 2.2. α M (ad) = +: By Oservation 2.3 we infer α M (a) = +. Sine ret(a) ret(ad), we onlude α M (ad) = +, whih onludes this ase. Therefore, α M gives a monotone orientation sequene for eah 4-tuple, and hene is a 3-signotope. Moreover, y the argument aove, it is easy to see that it never gives the monotone sequene (,,, +) or (+,,, ), and thus it is a restrited sulass of 3-signotopes. Felsner and Weil [28] assoiate an r-signotope α on [n] with a direted graph whose verties are the (r 1)-element susets of [n], and whose edges are ( ) [n] α {P i α P j P, 1 i < j r, α(p ) = +} r {P j α P i P ( ) [n], 1 i < j r, α(p ) = }. r They prove the following lemma. Lemma 7 (Felsner and Weil [28]). For an r-signotope α on [n] the graph with verties ( ) [n] r 1 and edges α is ayli. For the 3-signotope α M, the edge set αm of this assoiated direted graph onsists of the edges of D M with their diretions reversed. Therefore, α M eing a 3-signotope (Theorem 3), together with Lemma 7, implies that D M is ayli. Hene our orientation forms a partial order on the taleau entries. Lemma 6 guarantees that if a taleau agrees with our orientation, then it is alaned and has the desired skeleton. As a result, any taleau that realizes this partial order would e a alaned taleau with the desired skeleton. This onludes Theorem 2. A 3-signotope is realizale as an ordered generalized onfiguration of points (y Theorem 7 of Felsner and Weil [28] and the onept of duality etween pseudoline arrangements and generalized onfigurations of points). By ordered here we mean that in some diretion the projetion of the points (of the generalized onfiguration) are ordered from 1 to n. However, not all 3-signotopes are realizale as ordered point sets. Our orientation is a strit sulass of 3-signotopes. It exludes the monotone orientation sequenes (,,, +) and (+,,, ), whih puts more onstraints on the point set realizing it (that is, it forids ertain sustrutures in any realization). This may help prove realizaility. Felsner and Weil give an astrat ominatorial proof that the graphs assoiated with signotopes are ayli in general. A deeper understanding of the impliations of the persistent property on the slope ordering has the potential to determine additional onstraints that may aid in realization. In the next setion, we give an alternate self-ontained proof to larify these impliations and explain in more detail how the persistent property prevents yles. JoCG 6(1), ,