Upper Bounds on the Spanning Ratio of Constrained Theta-Graphs

Transcription

1 Upper Bonds on the Spanning Ratio of Constrained Theta-Graphs Prosenjit Bose and André van Renssen School of Compter Science, Carleton University, Ottaa, Canada. Abstract. We present tight pper and loer bonds on the spanning ratio of a large family of constrained -graphs. We sho that constrained -graphs ith and integer) cones have a tight spanning ratio of 1 + sin/), here is π/4 + ). We also present improved pper bonds on the spanning ratio of the other families of constrained -graphs. 1 Introdction A geometric graph G is a graph hose vertices are points in the plane and hose edges are line segments beteen pairs of points. Every edge is eighted by the Eclidean distance beteen its endpoints. The distance beteen to vertices and v in G, denoted by d G, v), is defined as the sm of the eights of the edges along the shortest path beteen and v in G. A sbgraph H of G is a t-spanner of G for t 1) if for each pair of vertices and v, d H, v) t d G, v). The smallest vale t for hich H is a t-spanner is the spanning ratio or stretch factor. The graph G is referred to as the nderlying graph of H. The spanning properties of varios geometric graphs have been stdied extensively in the literatre see [4,9] for a comprehensive overvie of the topic). We loo at a specific type of geometric spanner: -graphs. Introdced independently by Clarson [6] and Keil [8], -graphs partition the plane arond each vertex into m disjoint cones, each having apertre = π/m. The m -graph is constrcted by, for each cone of each vertex, connecting to the vertex v hose projection along the bisector of the cone is closest. Rppert and Seidel [10] shoed that the spanning ratio of these graphs is at most 1/1 sin/)), hen < π/3, i.e. there are at least seven cones. Recent reslts inclde a tight spanning ratio of 1 + sin/) for -graphs ith 4 + cones [1], here 1 and integer, and improved pper bonds for the other three families of -graphs [5]. Most of the research, hoever, has focsed on constrcting spanners here the nderlying graph is the complete Eclidean geometric graph. We stdy this problem in a more general setting ith the introdction of line segment constraints. Specifically, let P be a set of points in the plane and let S be a set Research spported in part by NSERC and Carleton University s President s 010 Doctoral Felloship.

2 Prosenjit Bose and André van Renssen of line segments beteen to vertices in P, called constraints. The set of constraints is planar, i.e. no to constraints intersect properly. To vertices and v can see each other if and only if either the line segment v does not properly intersect any constraint or v is itself a constraint. If to vertices and v can see each other, the line segment v is a visibility edge. The visibility graph of P ith respect to a set of constraints S, denoted VisP, S), has P as vertex set and all visibility edges as edge set. In other ords, it is the complete graph on P mins all edges that properly intersect one or more constraints in S. This setting has been stdied extensively ithin the context of motion planning amid obstacles. Clarson [6] as one of the first to stdy this problem and shoed ho to constrct a linear-sized 1+ɛ)-spanner of VisP, S). Sbseqently, Das [7] shoed ho to constrct a spanner of VisP, S) ith constant spanning ratio and constant degree. The Constrained Delanay Trianglation as shon to be a.4-spanner of VisP, S) [3]. Recently, it as also shon that the constrained 6 -graph is a -spanner of VisP, S) []. In this paper, e generalize the recent reslts on nconstrained -graphs to the constrained setting. There are to main obstacles that differentiate this or from previos reslts. First, the main difficlty ith the constrained setting is that indction cannot be applied directly, as the destination need not be visible from the vertex closest to the sorce see Figre 5, here is not visible from v 0, the vertex closest to ). Second, hen the graph does not have 4 + cones, the cones do not line p as nicely as in [], maing it more difficlt to apply indction. In this paper, e overcome these to difficlties and sho that constrained -graphs ith 4 + cones have a spanning ratio of at most 1+ sin/), here is π/4 + ). Since the loer bonds of the nconstrained -graphs carry over to the constrained setting, this shos that this spanning ratio is tight. We also sho that constrained -graphs ith cones have a spanning ratio of at most 1 + sin/)/cos/) sin/)), here is π/4 + 4). Finally, e sho that constrained -graphs ith 4 +3 or 4 +5 cones have a spanning ratio of at most cos/4)/cos/) sin3/4)), here is π/4 +3) or π/4 +5). Preliminaries We define a cone C to be the region in the plane beteen to rays originating from a vertex referred to as the apex of the cone. When constrcting a constrained) 4+x) -graph, for each vertex consider the rays originating from ith the angle beteen consective rays being = π/4 + x), here 1 and integer and x {, 3, 4, 5}. Each pair of consective rays defines a cone. The cones are oriented sch that the bisector of some cone coincides ith the vertical halfline throgh that lies above. Let this cone be C 0 of and nmber the cones in clocise order arond. The cones arond the other vertices have the same orientation as the ones arond. We rite Ci to indicate the i-th cone of a vertex. For ease of exposition, e only consider point sets in general position: no to points lie on a line parallel to one of the rays that define the cones, no to points lie on a line perpendiclar to the bisector of a cone, and no three points are collinear.

3 Bonds on Constrained Theta-Graphs 3 Let vertex be an endpoint of a constraint c and let the other endpoint v lie in cone Ci. The lines throgh all sch constraints c split C i into several sbcones. We se Ci,j to denote the j-th sbcone of C i. When a constraint c =, v) splits a cone of into to sbcones, e define v to lie in both of these sbcones. We consider a cone that is not split to be a single sbcone. We no introdce the constrained 4+x) -graph: for each sbcone C i,j of each vertex, add an edge from to the closest vertex in that sbcone that can see, here distance is measred along the bisector of the original cone not the sbcone). More formally, e add an edge beteen to vertices and v if v can see, v Ci,j, and for all points C i,j that can see, v, here v and denote the projection of v and on the bisector of Ci and xy denotes the length of the line segment beteen to points x and y. Note that or assmption of general position implies that each vertex adds at most one edge for each of its sbcones. Given a vertex in the cone C i of vertex, e define the canonical triangle T to be the triangle defined by the borders of Ci and m the line throgh perpendiclar to the bisector of Ci. Note that sbcones do not define α canonical triangles. We se m to denote the midpoint of the side of T opposing and α to denote the nsigned angle beteen and m see Figre 1). Note that for any pair of vertices and, there exist to canonical triangles: T and T. We say that a region is empty if it does not contain any vertex of P. Fig. 1. The canonical triangle T 3 Some Usefl Lemmas In this section, e list a nmber of lemmas that are sed hen bonding the spanning ratio of the varios graphs. Note that these lemmas are not ne, as they are already sed in [,5], thogh some are expanded to or for all for families of constrained -graphs. We start ith a nice v property of visibility graphs from []. Lemma 1. Let, v, and be three arbitrary points in the plane sch that and v are visibility edges and is not the endpoint of a constraint intersecting the interior of triangle v. Then there exists a convex chain of visibility edges from to v in triangle v, sch that the polygon defined by, v and the convex chain is empty and does not contain any constraints. x Fig.. The convex chain beteen vertices and v, here thic lines are visibility edges y

4 4 Prosenjit Bose and André van Renssen Next, e se to lemmas from [5] to bond the length of certain line segments. Note that Lemma is extended sch that it also holds for the constrained 4+) -graph. We se xyz to denote the smaller angle beteen line segments xy and yz. Lemma. Let, v and be three vertices in the 4+x) -graph, x {, 3, 4, 5}, sch that C0 and v T, to the left of. Let a be the intersection of the side of T opposite and the left bondary of C0 v. Let Ci v denote the cone of v that contains and let c and d be the pper and loer corner of T v. If 1 i 1, or i = and c d, then max { vc + c, vd + d } va + a and max { c, d } a. a c C v i d v a y β γ z v Fig. 3. The sitation here e apply Lemma Fig. 4. The sitation here e apply Lemma 3 Lemma 3. Let, v and be three vertices in the 4+x) -graph, x {, 3, 4, 5}, sch that C0, v T to the left of, and C0 v. Let a be the intersection of the side of T opposite and the line throgh v parallel to the left bondary of T. Let y and z be the corners of T v opposite to v. Let β = av and let γ be the nsigned angle beteen v and the bisector of T v. Let c be a positive cos γ sin β constant. If c cos β) sin +γ), then vp + c p va + c a, here p is y if y z and z if y < z. 4 Constrained 4+) -Graph In this section e prove that the constrained 4+) -graph has spanning ratio at most 1 + sin/). Since this is also a loer bond [1], this proves that this spanning ratio is tight.

5 Bonds on Constrained Theta-Graphs 5 Theorem 1. Let and be to vertices in the plane sch that can see. Let m be the midpoint of the side of T opposing and let α be the nsigned angle beteen and m. There exists a path connecting and in the constrained 4+) -graph of length at most ) ) 1 + sin ) cos ) cos α + sin α. Proof. We assme ithot loss of generality that C 0. We prove the theorem by indction on the area of T. Formally, e perform indction on the ran, hen ordered by area, of the triangles T xy for all pairs of vertices x and y that can see each other. Let a and b be the pper left and right corner of T, and let A and B be the triangles a and b see Figre 5). Or indctive hypothesis is the folloing, here δ, ) denotes the length of the shortest path from to in the constrained 4+) -graph: If A is empty, then δ, ) b + b. If B is empty, then δ, ) a + a. If neither A nor B is empty, then δ, ) max{ a + a, b + b }. We first sho that this indction hypothesis implies the theorem: m = cos α, m = sin α, am = bm = cos α tan/), and a = b = cos α/ cos/). Ths the indction hypothesis gives that δ, ) is at most 1 + sin/))/ cos/)) cos α + sin α). Base case: T has ran 1. Since the triangle is a smallest triangle, is the closest vertex to in that cone. Hence the edge, ) is part of the constrained 4+) -graph, and δ, ) =. From the triangle ineqality, e have min{ a + a, b + b }, so the indction hypothesis holds. Indction step: We assme that the indction hypothesis holds for all pairs of vertices that can see each other and have a canonical triangle hose area is smaller than the area of T. If, ) is an edge in the constrained 4+) -graph, the indction hypothesis follos a b by the same argment as in the base case. If there is no edge beteen and, let v 0 be the vertex closest to in the sbcone of that contains, and let a 0 and v be the pper left and right corner of b 0 T v0 see Figre 5). By definition, δ, ) v 0 + δv 0, ), and by the triangle ineqality, v 0 min{ a 0 + a 0 v 0, b 0 + b 0 v 0 }. We assme ithot loss of generality that v 0 lies to the left of, hich means that A is not empty. Since and v 0 are visibility edges, by applying Lemma 1 to triangle v 0, a convex chain v 0,..., v l = of visibility edges v 0 v 1 a 0 b 0 Fig. 5. A convex chain from v 0 to

6 6 Prosenjit Bose and André van Renssen connecting v 0 and exists see Figre 5). Note that, since v 0 is the closest visible vertex to, every vertex along the convex chain lies above the horizontal line throgh v 0. We no loo at to consective vertices v j 1 and v j along the convex chain. There are for types of configrations see Figre 6): i) v j C vj 1, ii) v j C vj 1 i here 1 i <, iii) v j C vj 1 0 and v j lies to the right of or has the same x-coordinate as v j 1, iv) v j C vj 1 0 and v j lies to the left of v j 1. By convexity, the direction of v j v j+1 is rotating conterclocise for increasing j. Ths, these configrations occr in the order Type i), Type ii), Type iii), Type iv) along the convex chain from v 0 to. We bond δv j 1, v j ) as follos: Type i): If v j C vj 1, let a j and b j be the pper and loer left corner of T vjv j 1 and let B j = v j 1 b j v j. Note that since v j C vj 1, a j is also the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. Triangle B j lies beteen the convex chain and, so it mst be empty. Since v j can see v j 1 and T vjv j 1 has smaller area than T, the indction hypothesis gives that δv j 1, v j ) is at most v j 1 a j + a j v j. a j v j c a j v j b j a j a j v j d v j b j v j 1 b j v j 1 v j 1 v j 1 i) ii) iii) iv) Fig. 6. The for types of configrations Type ii): If v j C vj 1 i here 1 i <, let c and d be the pper and loer right corner of T vj 1v j. Let a j be the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. Since v j can see v j 1 and T vj 1v j has smaller area than T, the indction hypothesis gives that δv j 1, v j ) is at most max{ v j 1 c + cv j, v j 1 d + dv j }. Since v j C vj 1 i here 1 i <, e can apply Lemma here v,, and a from Lemma are v j 1, v j, and a j ), hich gives s that max{ v j 1 c + cv j, v j 1 d + dv j } v j 1 a j + a j v j. Type iii): If v j C vj 1 0 and v j lies to the right of or has the same x- coordinate as v j 1, let a j and b j be the left and right corner of T vj 1v j and let A j = v j 1 a j v j and B j = v j 1 b j v j. Since v j can see v j 1 and T vj 1v j has smaller area than T, e can apply the indction hypothesis. Regardless of hether A j and B j are empty or not, δv j 1, v j ) is at most max{ v j 1 a j + a j v j, v j 1 b j + b j v j }. Since v j lies to the right of or has the same x-coordinate as v j 1, e no that v j 1 a j + a j v j v j 1 b j + b j v j, so δv j 1, v j ) is at most v j 1 a j + a j v j. Type iv): If v j C vj 1 0 and v j lies to the left of v j 1, let a j and b j be the left and right corner of T vj 1v j and let A j = v j 1 a j v j and B j = v j 1 b j v j. Since v j can see v j 1 and T vj 1v j has smaller area than T, e can apply the

7 Bonds on Constrained Theta-Graphs 7 b b b v j a v j v j b Fig. 7. Visalization of the paths thic lines) in the ineqalities of case c) indction hypothesis. Ths, if B j is empty, δv j 1, v j ) is at most v j 1 a j + a j v j and if B j is not empty, δv j 1, v j ) is at most v j 1 b j + b j v j. To complete the proof, e consider three cases: a) a π/, b) a > π/ and B is empty, c) a > π/ and B is not empty. Case a): If a π/, the convex chain cannot contain any Type iv) configrations: for Type iv) configrations to occr, v j needs to lie to the left of v j 1. Hoever, by constrction, v j lies on or to the right of the line throgh v j 1 and. Hence, since av j 1 < a π/, v j lies to the right of or has the same x-coordinate as v j 1. We can no bond δ, ) by sing these bonds: δ, ) v 0 + l j=1 δv j 1, v j ) a 0 + a 0 v 0 + l j=1 v j 1a j + a j v j ) = a + a. Case b): If a > π/ and B is empty, the convex chain can contain Type iv) configrations. Hoever, since B is empty and the area beteen the convex chain and is empty by Lemma 1), all B j are also empty. Using the compted bonds on the lengths of the paths beteen the points along the convex chain, e can bond δ, ) as in the previos case. Case c): If a > π/ and B is not empty, the convex chain can contain Type iv) configrations and since B is not empty, the triangles B j need not be empty. Recall that v 0 lies in A, hence neither A nor B are empty. Therefore, it sffices to prove that δ, ) max{ a + a, b + b } = b + b. Let T vj v j +1 be the first Type iv) configration along the convex chain if it has any), let a and b be the pper left and right corner of T vj, and let b be the pper right corner of T vj. We no have that δ, ) v 0 + l j=1 δv j 1, v j ) a + a v j + v j b + b b + b see Figre 7). Since 1 + sin/))/ cos/)) cos α + sin α is increasing for α [0, /], for π/3, it is maximized hen α = /, and e obtain the folloing corollary: Corollary 1. The constrained 4+) -graph is a 1 + sin )) -spanner of VisP, S).

8 8 Prosenjit Bose and André van Renssen 5 Generic Frameor for the Spanning Proof Next, e modify the spanning proof from the previos section and provide a generic frameor for the spanning proof for the other three families of -graphs. After providing this frameor, e fill in the blans for the individal families. Theorem. Let and be to vertices in the plane sch that can see. Let m be the midpoint of the side of T opposing and let α be the nsigned angle beteen and m. There exists a path connecting and in the constrained 4+x) -graph of length at most cos α cos ) + cos α tan ) ) ) + sin α c, here c 1 is a constant that depends on x {3, 4, 5}. For the constrained 4+4) -graph, c eqals 1/cos/) sin/)) and for the constrained 4+3) - graph and 4+5) -graph, c eqals cos/4)/cos/) sin3/4)). Proof. We prove the theorem by indction on the area of T. Formally, e perform indction on the ran, hen ordered by area, of the triangles T xy for all pairs of vertices x and y that can see each other. We assme ithot loss of generality that C 0. Let a and b be the pper left and right corner of T see Figre 5). Or indctive hypothesis is the folloing, here δ, ) denotes the length of the shortest path from to in the constrained 4+x) -graph: δ, ) max{ a + a c, b + b c}. We first sho that this indction hypothesis implies the theorem. Basic trigonometry gives s the folloing eqalities: m = cos α, m = sin α, am = bm = cos α tan/), and a = b = cos α/ cos/). Ths the indction hypothesis gives that δ, ) is at most cos α/ cos/) + cos α tan/) + sin α) c). Base case: T has ran 1. Since the triangle is a smallest triangle, is the closest vertex to in that cone. Hence the edge, ) is part of the constrained 4+x) -graph, and δ, ) =. From the triangle ineqality and the fact that c 1, e have min{ a + a c, b + b c}, so the indction hypothesis holds. Indction step: We assme that the indction hypothesis holds for all pairs of vertices that can see each other and have a canonical triangle hose area is smaller than the area of T. If, ) is an edge in the constrained 4+x) -graph, the indction hypothesis follos by the same argment as in the base case. If there is no edge beteen and, let v 0 be the vertex closest to in the sbcone of that contains, and let a 0 and b 0 be the pper left and right corner of T v0 see Figre 5). By definition, δ, ) v 0 + δv 0, ), and by the triangle ineqality, v 0 min{ a 0 + a 0 v 0, b 0 + b 0 v 0 }. We assme ithot loss of generality that v 0 lies to the left of.

9 Bonds on Constrained Theta-Graphs 9 Since and v 0 are visibility edges, by applying Lemma 1 to triangle v 0, a convex chain v 0,..., v l = of visibility edges connecting v 0 and exists see Figre 5). Note that, since v 0 is the closest visible vertex to, every vertex along the convex chain lies above the horizontal line throgh v 0. We no loo at to consective vertices v j 1 and v j along the convex chain. When v j C vj 1 0, let c and d be the pper and loer right corner of T vj 1v j. We distingish for types of configrations: i) v j C vj 1 i here i >, or i = and c > d, ii) v j C vj 1 i here 1 i 1, or i = and c d, iii) v j C vj 1 0 and v j lies to the right of or has the same x-coordinate as v j 1, iv) v j C vj 1 0 and v j lies to the left of v j 1. By convexity, the direction of v j v j+1 is rotating conterclocise for increasing j. Ths, these configrations occr in the order Type i), Type ii), Type iii), Type iv) along the convex chain from v 0 to. We bond δv j 1, v j ) as follos: Type i): v j C vj 1 i here i >, or i = and c > d. Since v j can see v j 1 and T vjv j 1 has smaller area than T, the indction hypothesis gives that δv j 1, v j ) is at most max{ v j 1 c + cv j c, v j 1 d + dv j c}. Let a j be the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. We aim to sho that max{ v j 1 c + cv j c, v j 1 d + dv j c} v j 1 a j + a j v j c. We se Lemma 3 to do this. Hoever, since the precise application of this lemma depends on the family of -graphs and determines the vale of c, this case is discssed in the spanning proofs of the three families. Type ii): v j C vj 1 i here 1 i 1, or i = and c d. Since v j can see v j 1 and T vjv j 1 has smaller area than T, the indction hypothesis gives that δv j 1, v j ) is at most max{ v j 1 c + cv j c, v j 1 d + dv j c}. Let a j be the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. Since v j C vj 1 i here 1 i 1, or i = and c d, e can apply Lemma in this case here v,, and a from Lemma are v j 1, v j, and a j ) and e get that max{ v j 1 c + cv j, v j 1 d + dv j } v j 1 a j + a j v j and max{ cv j, dv j } a j v j. Since c 1, this implies that max{ v j 1 c + cv j c, v j 1 d + dv j c} v j 1 a j + a j v j c. Type iii): If v j C vj 1 0 and v j lies to the right of or has the same x- coordinate as v j 1, let a j and b j be the left and right corner of T vj 1v j. Since v j can see v j 1 and T vj 1v j has smaller area than T, e can apply the indction hypothesis. Ths, since v j lies to the right of or has the same x-coordinate as v j 1, δv j 1, v j ) is at most v j 1 a j + a j v j c. Type iv): If v j C vj 1 0 and v j lies to the left of v j 1, let a j and b j be the left and right corner of T vj 1v j. Since v j can see v j 1 and T vj 1v j has smaller area than T, e can apply the indction hypothesis. Ths, since v j lies to the left of v j 1, δv j 1, v j ) is at most v j 1 b j + b j v j c. To complete the proof, e consider to cases: a) a π, b) a > π. Case a): We need to prove that δ, ) max{ a + a, b + b } = a + a. We first sho that the convex chain cannot contain any Type iv) configrations: for Type iv) configrations to occr, v j needs to lie to the left of v j 1. Hoever, by constrction, v j lies on or to the right of the line throgh v j 1 and. Hence, since av j 1 < a π/, v j lies to the right of v j 1. We can

10 10 Prosenjit Bose and André van Renssen no bond δ, ) by sing these bonds: δ, ) v 0 + l j=1 δv j 1, v j ) a 0 + a 0 v 0 + l j=1 v j 1a j + a j v j c) a + a c. Case b): If a > π/, the convex chain can contain Type iv) configrations. We need to prove that δ, ) max{ a + a, b + b } = b + b. Let T vj v j +1 be the first Type iv) configration along the convex chain if it has any), let a and b be the pper left and right corner of T vj, and let b be the pper right corner of T vj. We no have that δ, ) v 0 + l j=1 δv j 1, v j ) a + a v j c + v j b + b c b + b c see Figre 7). 6 The Constrained 4+4) -Graph In this section e complete the proof of Theorem for the constrained 4+4) - graph. Theorem 3. Let and be to vertices in the plane sch that can see. Let m be the midpoint of the side of T opposite and let α be the nsigned angle beteen and m. There exists a path connecting and in the constrained 4+4) -graph of length at most cos α cos ) + cos α tan ) ) + sin α cos ) sin ). Proof. We apply Theorem sing c = 1/cos/) sin/)). The assmptions made in Theorem still apply. It remains to sho that for the Type i) configrations, e have that max{ v j 1 c + cv j c, v j 1 d + dv j c} v j 1 a j + a j v j c, here c and d are the pper and loer right corner of T vj 1v j and a j is the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. We distingish to cases: a) v j C vj 1 and c > d, b) v j C vj Let β be a j v j v j 1 and let γ be the angle beteen v j v j 1 and the bisector of T vj 1v j. Case a): When v j C vj 1 and c > d, the indction hypothesis for T vj 1v j gives δv j 1, v j ) v j 1 c + cv j c. We note that γ = β. Hence Lemma 3 gives that the ineqality holds hen c cos β) sin β)/cos/ β) sin3/ β)). As this fnction is decreasing in β for / β, it is maximized hen β eqals /. Hence c needs to be at least cos/) sin/))/1 sin ), hich can be reritten to 1/cos/) sin/)). Case b): When v j C vj 1 +1, v j lies above the bisector of T vj 1v j and the indction hypothesis for T vj 1v j gives δv j 1, v j ) v j d + dv j 1 c. We note that γ = β. Hence Lemma 3 gives that the ineqality holds hen c cos β sin β)/cos/ β) sin/ + β)). As this fnction is decreasing in β for 0 β /, it is maximized hen β eqals 0. Hence c needs to be at least 1/cos/) sin/)). Since cos α/ cos/) + cos α tan/) + sin α)/cos/) sin/)) is increasing for α [0, /], for π/4, it is maximized hen α = /, and e obtain the folloing corollary:

11 Corollary. The constrained 4+4) -graph is a of VisP, S). Bonds on Constrained Theta-Graphs 11 ) sin 1 + ) -spanner cos ) sin ) 7 The Constrained 4+3) -Graph and 4+5) -Graph In this section e complete the proof of Theorem for the constrained 4+3) - graph and 4+5) -graph. Theorem 4. Let and be to vertices in the plane sch that can see. Let m be the midpoint of the side of T opposite and let α be the nsigned angle beteen and m. There exists a path connecting and in the constrained 4+3) -graph of length at most ) ) ) + sin α cos 4) cos α cos α tan cos ) + cos ) sin 3 ) 4. Proof. We apply Theorem sing c = cos/4)/cos/) sin3/4)). The assmptions made in Theorem still apply. It remains to sho that for the Type i) configrations, e have that max{ v j 1 c + cv j c, v j 1 d + dv j c} v j 1 a j + a j v j c, here c and d are the pper and loer right corner of T vj 1v j and a j is the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. We distingish to cases: a) v j C vj 1 and c > d, b) v j C vj Let β be a j v j v j 1 and let γ be the angle beteen v j v j 1 and the bisector of T vj 1v j. Case a): When v j C vj 1 and c > d, the indction hypothesis for T vj 1v j gives δv j 1, v j ) v j 1 c + cv j c. We note that γ = 3/4 β. Hence Lemma 3 gives that the ineqality holds hen c cos3/4 β) sin β)/cos/ β) sin5/4 β)). As this fnction is decreasing in β for /4 β 3/4, it is maximized hen β eqals /4. Hence c needs to be at least cos/) sin/4))/cos/4) sin ), hich is eqal to cos/4)/cos/) sin3/4)). Case b): When v j C vj 1 +1, v j lies above the bisector of T vj 1v j and the indction hypothesis for T vj 1v j gives δv j 1, v j ) v j d + dv j 1 c. We note that γ = /4+β. Hence Lemma 3 gives that the ineqality holds hen c cos/4+ β) sin β)/cos/ β) sin3/4+β)), hich is eqal to cos/4)/cos/) sin3/4)). Theorem 5. Let and be to vertices in the plane sch that can see. Let m be the midpoint of the side of T opposite and let α be the nsigned angle beteen and m. There exists a path connecting and in the constrained 4+5) -graph of length at most ) ) ) + sin α cos 4) cos α cos α tan cos ) + cos ) sin 3 ) 4.

12 1 Prosenjit Bose and André van Renssen Proof. We apply Theorem sing c = cos/4)/cos/) sin3/4)). The assmptions made in Theorem still apply. It remains to sho that for the Type i) configrations, e have that max{ v j 1 c + cv j c, v j 1 d + dv j c} v j 1 a j + a j v j c, here c and d are the pper and loer right corner of T vj 1v j and a j is the intersection of the left bondary of C vj 1 0 and the horizontal line throgh v j. We distingish to cases: a) v j C vj 1 and c > d, b) v j C vj Let β be a j v j v j 1 and let γ be the angle beteen v j v j 1 and the bisector of T vj 1v j. Case a): When v j C vj 1 and c > d, the indction hypothesis for T vj 1v j gives δv j 1, v j ) v j 1 c + cv j c. We note that γ = 5/4 β. Hence Lemma 3 gives that the ineqality holds hen c cos5/4 β) sin β)/cos/ β) sin5/4 β)). As this fnction is decreasing in β for 3/4 β 5/4, it is maximized hen β eqals 3/4. Hence c needs to be at least cos/) sin3/4))/cos/4) sin ), hich is less than cos/4)/cos/) sin3/4)). Case b): When v j C vj 1 +1, the indction hypothesis for T v gives δv j 1, v j ) max{ v j 1 c + cv j c, v j 1 d + dv j c}. If δv j 1, v j ) v j 1 c + cv j c, e note that γ = /4 β. Hence Lemma 3 gives that the ineqality holds hen c cos/4 β) sin β)/cos/ β) sin3/4 β)). As this fnction is decreasing in β for 0 β /4, it is maximized hen β eqals 0. Hence c needs to be at least cos/4)/cos/) sin3/4)). If δv j 1, v j ) v j 1 d + dv j c, e note that γ = /4 + β. Hence Lemma 3 gives that the ineqality holds hen c cosβ /4) sin β)/cos/ β) sin/4 + β)), hich is eqal to cos/4)/cos/) sin3/4)). When looing at to vertices and in the constrained 4+3) -graph and 4+5) -graph, e notice that hen the angle beteen and the bisector of T is α, the angle beteen and the bisector of T is / α. Hence the orst case spanning ratio becomes the minimm of the spanning ratio hen looing at T and the spanning ratio hen looing at T. Theorem 6. The constrained 4+3) -graph and 4+5) -graph are cos 4 ) -spanners of VisP, S). cos ) sin 3 4 ) Proof. The spanning ratio of the constrained 4+3) -graph and 4+5) -graph is at most: cos α min cos ) + cos α tan )+sin α) cos 4 ), cos ) sin 3 4 ) cos α) + cos α) tan )+sin α)) cos 4 ) cos ) cos ) sin 3 4 ) Since cos α/ cos/)+cos α tan/)+sin α) c is increasing for α [0, /], for π/7, the minimm of these to fnctions is maximized hen the to fnctions are eqal, i.e. hen α = /4. Ths the constrained 4+3) -graph and

13 Bonds on Constrained Theta-Graphs ) -graph has spanning ratio at most: cos ) 4 cos ) cos 4 tan ) + sin )) 4 cos ) + 4) cos ) sin 3 ) = 4 cos 4) cos ) cos ) cos ) sin 3 4 )) References 1. P. Bose, J.-L. De Carfel, P. Morin, A. van Renssen, and S. Verdonschot. Optimal bonds on theta-graphs: More is not alays better. In Proceedings of the 4th Canadian Conference on Comptational Geometry CCCG 01), pages , 01.. P. Bose, R. Fagerberg, A. van Renssen, and S. Verdonschot. On plane constrained bonded-degree spanners. In Proceedings of the 10th Latin American Symposim on Theoretical Informatics LATIN 01), volme 756 of Lectre Notes in Compter Science, pages 85 96, P. Bose and J. M. Keil. On the stretch factor of the constrained Delanay trianglation. In Proceedings of the 3rd International Symposim on Voronoi Diagrams in Science and Engineering ISVD 006), pages 5 31, P. Bose and M. Smid. On plane geometric spanners: A srvey and open problems. In Comptational Geometry: Theory and Applications CGTA), accepted, P. Bose, A. van Renssen, and S. Verdonschot. On the spanning ratio of thetagraphs. In Proceedings of the 13th Worshop on Algorithms and Data Strctres WADS 013), volme 8037 of Lectre Notes in Compter Science, pages , K. Clarson. Approximation algorithms for shortest path motion planning. In Proceedings of the 19th Annal ACM Symposim on Theory of Compting STOC 1987), pages 56 65, G. Das. The visibility graph contains a bonded-degree spanner. In Proceedings of the 9th Canadian Conference on Comptational Geometry CCCG 1997), pages 70 75, J. Keil. Approximating the complete Eclidean graph. In Proceedings of the 1st Scandinavian Worshop on Algorithm Theory SWAT 1988), pages 08 13, G. Narasimhan and M. Smid. Geometric Spanner Netors. Cambridge University Press, J. Rppert and R. Seidel. Approximating the d-dimensional complete Eclidean graph. In Proceedings of the 3rd Canadian Conference on Comptational Geometry CCCG 1991), pages 07 10, 1991.