Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
Nicolas Bonichon, Prosenjit Bose, Paz Carmi, Irina Kostitsyna, Anna Lubiw, Sander Verdonschot
GGabriel Triangulations and Angle-MonotoneGraphs: Local Routing and Recognition
Nicolas Bonichon , Prosenjit Bose , Paz Carmi ,Irina Kostitsyna , Anna Lubiw , and Sander Verdonschot LaBRI, Univ. Bordeaux, France. [email protected] School of Computer Science, Carleton University, Ottawa, Canada. [email protected] Department of Computer Science, Ben-Gurion University of the Negev, Israel. [email protected] Universit´e Libre de Bruxelles. [email protected] David R. Cheriton School of Computer Science, University of Waterloo, Canada. [email protected] School of Electrical Engineering and Computer Science, University of Ottawa,Ottawa, Canada. [email protected]
Abstract.
A geometric graph is angle-monotone if every pair of ver-tices has a path between them that—after some rotation—is x - and y -monotone. Angle-monotone graphs are √ generalized angle-monotone —specifically,we prove that the half- θ -graph is generalized angle-monotone. We give alocal routing algorithm for Gabriel triangulations that finds a path fromany vertex s to any vertex t whose length is within 1 + √ s to t . Finally, we prove some lower bounds andlimits on local routing algorithms on Gabriel triangulations. A geometric graph has vertices that are points in the plane, and edges that aredrawn as straight-line segments, with the weight of an edge being its Euclideanlength. A geometric graph need not be planar. Geometric graphs that haverelatively short paths are relevant in many applications for routing and networkdesign, and have been a subject of intense research. A main scenario is that weare given a point set and must construct a sparse geometric graph on that pointset with good shortest path properties.If the shortest path between every pair of points has length at most t timesthe Euclidean distance between the points, then the geometric graph is calleda t - spanner , and the minimum such t is called the spanning ratio . Since theirintroduction by Paul Chew in 1986 [10], spanners have been heavily studied [18]. a r X i v : . [ c s . C G ] A ug esides the existence of short paths, another issue is routing —how to findshort paths in a geometric graph. One goal is to find paths using local routing where the path is found one vertex at a time using only local information aboutthe neighbours of the current vertex plus the coordinates of the destination.A main example of such a method is greedy routing : from the current vertex u take any edge to a vertex v that is closer (in Euclidean distance) to thedestination than u is. The geometric graphs for which greedy routing succeedsin finding a path are called greedy drawings . These have received considerableattention because of their potential ability to replace routing tables for networkrouting, and because of the noted conjecture of Papadimitriou and Ratajczak [19](proved in [5, 16]) that every 3-connected planar graph has a greedy drawing.One drawback is that a path found by greedy routing may be very long comparedto the Euclidean distance between the endpoints. Of course this is inevitable ifthe geometric graph has large spanning ratio.When a geometric graph is a t -spanner, we can ideally hope for a local routingalgorithm that finds a path whose length is at most k times the Euclideandistance between the endpoints, for some k , where, of necessity, k ≥ t . Themaximum ratio, k , of path length to Euclidean distance is called the routing ratio .For example, the Delaunay triangulation, which is a t -spanner for t ≤ .
998 [21],permits local routing with routing ratio k ≤ .
90 [7]. It is an open questionwhether the spanning ratio and routing ratio are equal, though there is a provablegap for L -Delaunay triangulations [7] and TD-Delaunay triangulations [9]. Other “good” paths.
Recently, a number of other notions of “good” pathsin geometric graphs have been investigated. Alamdari et al. [2] introduced self-approaching graphs, where any two vertices s and t are joined by a self-approachingpath —a path such that a point moving continuously along the path from s to any intermediate destination r on the path always gets closer to r in Eu-clidean distance. In an increasing-chord graph, this property also holds for thereverse path from t to s . The self-approaching path property is stronger than thegreedy path property in two ways: it applies to every intermediate destination r , and it requires that continuous motion (not just the vertices) along the pathto r always gets closer to r . The significance of the stronger property is thatself-approaching and increasing-chord graphs have bounded spanning ratios of5.333 [15] and 2.094 [20], respectively. An important characterization is that apath is self-approaching if and only if at each point on the path, there is a 90 ◦ wedge that contains the rest of the path [15].Angelini et al. [4] introduced monotone drawings , where any two vertices s and t are joined by a path that is monotone in some direction. This is a naturaldesirable property, but not enough to guarantee a bounded spanning ratio. Angle-monotone paths.
In this paper we explore properties of another classof geometric graphs with good path properties. These are the angle-monotonegraphs which were first introduced by Dehkordi, Frati, and Gudmundsson [12] asa tool to investigate increasing-chord graphs. (We note that Dehkordi et al. [12]did not give a name to their graph class.) polygonal path with vertices v , v , . . . , v n is β -monotone for some angle β if the vector of every edge ( v i , v i +1 ) lies in the closed 90 ◦ wedge between β − ◦ and β + 45 ◦ . (In the terminology of Dehkordi et al. [12] this is a θ -path .)In particular, an x - y -monotone path (where x and y coordinates are both non-decreasing along the path) is a β -monotone path for β = 45 ◦ (measured from thepositive x -axis). A path is angle-monotone if there is some angle β for which it is β -monotone. To visualize this, note that a path is angle-monotone if and only if itcan be rotated to be x - y -monotone. An angle-monotone path is a special case ofa self-approaching path where the wedges containing the rest of the path all havethe same orientation. See Figure 1. This implies that an angle-monotone pathis also angle-monotone when traversed in the other direction, and thus, has theincreasing-chord property. Observe that angle-monotone paths have spanningratio √ x - y -monotone paths do. s t s t β Fig. 1.
The difference between a self-approaching st path (left) with 90 ◦ wedges eachcontaining the rest of the path, and an angle-monotone path (right) where the 90 ◦ wedges all have the same orientation β . A geometric graph is angle-monotone if for every pair of vertices u , v , there isan angle-monotone path from u to v . Note that the angle β may be different fordifferent pairs u, v . Dehkhori et al. [12] introduced angle-monotone graphs, andproved that they include the class of Gabriel triangulations (triangulations withno obtuse angle). Their main goal was to prove that any set of n points in theplane has a planar increasing-chord graph with O ( n ) Steiner points and O ( n )edges. Given their result that Gabriel graphs are increasing chord, this followsfrom a result of Bern et al. [6] that any point set can be augmented with O ( n )points to a point set whose Delaunay triangulation is Gabriel.The notion of angle-monotone graphs can be generalized to wedges of angle γ different from 90 ◦ . (A precise definition is given below.) We call these angle-monotone graphs with width γ , or generalized angle-monotone graphs. For γ < ◦ , they still have bounded spanning ratios. Results.
The main themes we explore are: Which geometric graphs are angle-monotone? Can we create a sparse (generalized) angle-monotone graph on anygiven point set? Do angle-monotone graphs permit local routing?ur first main result is a polynomial time algorithm to test if a geometricgraph is angle-monotone. This is significant because it is not known whether in-creasing chord graphs can be recognized in polynomial time (or whether the prob-lem is NP-hard). Our algorithm extends to generalized angle-monotone graphsfor any width γ < ◦ .Our next result is that for any point set in the plane, there is a plane geometricgraph on that point set that is angle-monotone with width 120 ◦ . In particular,we prove that the half- θ -graph has this property. Width 90 ◦ cannot alwaysbe achieved because it would imply spanning ratio √ √ ≈ .
41. This is better than the best known routing ratio for Delaunaytriangulations of 5.90 [7]. Also, our algorithm is simpler. The algorithm succeeds,i.e. finds a path to the destination, for any triangulation, and we prove that thealgorithm has a bounded routing ratio for triangulations with maximum angleless than 120 ◦ . For Delaunay triangulations, we prove a lower bound on therouting ratio of 5.07, but leave as an open question whether the algorithm everdoes worse. Finally, we give some lower bounds on the routing ratio of localrouting algorithms on Gabriel triangulations, and we prove that no local routingalgorithm on Gabriel triangulations can find self-approaching paths.As is clear from this outline, we leave many interesting open questions, someof which are listed in the Conclusions section. Further Background.
The standard Delaunay triangulation is not self-ap-proaching in general [2], and therefore not angle-monotone.The
Gabriel graph of point set P is a graph in which for every edge ( u, v )the circle with diameter uv contains no points of P . A Gabriel graph that isa triangulation is called a Gabriel triangulation . Any Gabriel triangulation is aDelaunay triangulation. Observe that a triangulation is Gabriel if and only if ithas no obtuse angles. Not every point set has a Gabriel triangulation, e.g. threepoints forming an obtuse triangle.There are several results on constructing self-approaching/increasing-chordgraphs on a given set of points. Alamdari et al. [2] constructed an increasingchord network of linear size using Steiner points, and Dehkordi et al. [12] im-proved this to a plane network. It is an open question whether every point setadmits a plane increasing-chord graph without adding Steiner points. However,for the more restrictive case of angle-monotone graphs, the answer is no: anyangle-monotone graph has spanning ratio √ Preliminaries and Definitions.
A polygonal path with vertices v , v , . . . , v n is β -monotone with width γ for some angles β and γ with γ < ◦ if thevector of every edge ( v i , v i +1 ) lies in the closed wedge of angle γ between β − γ and β + γ . When we have no need to specify β , we say that the path is ngle-monotone with width γ , or “generalized angle-monotone”. A path that isgeneralized angle-monotone is a generalized self-approaching path [1] and thushas bounded spanning ratio depending on γ [1]. But in fact, we can do better: Observation 1 [proof in Appendix A] The spanning ratio of an angle-monotonepath with width γ < ◦ is at most / cos γ . A geometric graph is angle-monotone with width γ if for every pair of vertices u , v , there is an angle-monotone path with width γ from u to v . When we haveno need to specify γ , we say that the graph is “generalized angle-monotone”.Note that in an angle-monotone path (with width 90 ◦ ) the distances from v to later vertices form an increasing sequence. Furthermore, any β -monotonepath from u to v lies in a rectangle with u and v at opposite corners and withtwo sides at angles β ± ◦ , and the union of such rectangles over all β ∈ [0 , ◦ )forms the disc with diameter uv . (See Figure 5 in Appendix.) This implies: Lemma 1.
Any angle-monotone path from u to v lies inside the disc with di-ameter uv . In this section we give an O ( nm ) time algorithm to test if a geometric graphwith n vertices and m edges is angle-monotone. The idea is to look for angle-monotone paths from a node s to all other nodes, and then repeat over all choicesof s . For a given source vertex s , the algorithm explores nodes u in non-decreasingorder of their distance from s . At each vertex u we store information to captureall the possible angles β for which there is a β -monotone path from s to u . Weshow how to propagate this information along an edge from u to v .We begin with some notation. We will measure angles counterclockwise fromthe positive x -axis, modulo 360 ◦ . To any ordered pair u, v of vertices (points) ofour geometric graph we associate the vector v − u and we denote its angle by α ( u, v ). If S is a set of angles that lie within a wedge of angle less than 180 ◦ , thenwe define the minimum of S to be the most clockwise angle, and the maximum of S to be the most counter-clockwise angle. More formally, α is the minimum of S if for any other β ∈ S , β − α ∈ [0 , ◦ ), and similarly for maximum .Although there may be exponentially many angle-monotone paths from s to u , each such path has two extreme edges. More precisely, if P is an angle-monotone path from s to u , then the angles, α ( e ) , e ∈ P , lie in a 90 ◦ wedge, andso this set has a minimum and maximum that differ by at most 90 ◦ . We willstore a list of all such min-max pairs with vertex u . Each pair defines a wedgeof at most 90 ◦ . Since each pair is defined by two edges, there are at most O ( m )such pairs (though we will show below that we only need to store O ( m ) of them).The algorithm starts off by looking at every edge ( s, u ) and adding the pair( α ( s, u ) , α ( s, u )) to u ’s list. Then the algorithm explores vertices u (cid:54) = s in non-decreasing order of their distance from s . To explore vertex u , consider each edge( u, v ) and each pair ( α ( e ) , α ( f )) stored with u , and update the list of pairs forertex v as follows. If α ( u, v ) is within 90 ◦ of α ( e ) and within 90 ◦ of α ( f ) thenadd to v ’s list the pair (min { α ( u, v ) , α ( e ) } , max { α ( u, v ) , α ( f ) } ).If ever the algorithm tries to explore a vertex that has no pairs stored withit, then halt—the graph is not angle-monotone. To justify correctness we prove: Lemma 2.
When the algorithm has explored all the vertices closer to s than v ,then there exists an angle-monotone path from s to v with extreme edges e and f if and only if the pair ( α ( e ) , α ( f )) is in v ’s list.Proof. The proof is by induction on the distance from s to v .For the “only if” direction, let P be an angle-monotone path from s to v withextreme edges e and f , and let u be the penultimate vertex of P . The subpathof P from s to u is an angle-monotone path. Suppose its extreme edges are e (cid:48) and f (cid:48) where e = e (cid:48) or f = f (cid:48) or both. Now, u is closer to s so by induction thepair ( α ( e (cid:48) ) , α ( f (cid:48) )) is in u ’s list. Because P is angle-monotone, α ( u, v ) is within90 ◦ of α ( e (cid:48) ) and α ( f (cid:48) ). Thus the update step applies. During the update step weadd the angle α ( u, v ) to the pair ( α ( e (cid:48) ) , α ( f (cid:48) )), which gives the pair ( α ( e ) , α ( f )).Thus we add the pair ( α ( e ) , α ( f )) to v ’s list.For the “if” direction, suppose that the pair ( α ( e ) , α ( f )) is in v ’s list. Thispair was added to v ’s list because of an update from some vertex u closer to s applied to some pair ( α ( e (cid:48) ) , α ( f (cid:48) )) in u ’s list. By induction, there exists anangle-monotone path P from s to u with extreme edges e (cid:48) and f (cid:48) , and becausethe update is only performed when α ( u, v ) is within 90 ◦ degrees of α ( e (cid:48) ) and α ( f (cid:48) ) therefore the edge ( u, v ) can be added to P to produce an angle-monotonepath with extreme edges e and f . (cid:117)(cid:116) To improve the efficiency of the algorithm we observe that it is redundantto store at a vertex v a pair whose wedge contains the wedge of another pair.Therefore, we only need to store O ( m ) pairs at each vertex, at most one pairwhose first element is α ( e ) for each edge e . We can simply keep with each vertex v a vector indexed by edges e , in which we store the minimal pair ( α ( e ) , α ( f ))(if any) associated with v so far. Finally, observe that during the course ofthe algorithm, each edge ( u, v ) is handled once in an update step. With therefinement just mentioned, handling an edge costs O ( m ). Therefore the algorithmruns in time O ( m ) for a single choice of s , and in time O ( nm ) overall.The algorithm can be generalized to recognize angle-monotone graphs ofwidth γ for fixed γ < ◦ . It is no longer legitimate to explore vertices in orderof distance from s , since a generalized angle-monotone path will not necessarilyrespect this ordering. However, we can run the algorithm in phases, where phase i captures all the angle-monotone paths of width γ that start at s and have atmost i edges. Since no angle-monotone path can repeat a vertex, there are atmost n − n − u, v ) we update each pair ( α ( e ) , α ( f )) storedat u as follows. If α ( u, v ) is within γ of α ( e ) and within γ of α ( f ) then add to v ’s list the pair (min { α ( u, v ) , α ( e ) } , max { α ( u, v ) , α ( f ) } ). In this way, each of the n − O ( m ), so the total run-time of the algorithm over allchoices of s becomes O ( n m ). A Class of Generalized Angle-Monotone Graphs
In this section we show that every point set in the plane has a plane geometricgraph that is angle-monotone with width 120 ◦ . In particular, we will prove thatthe half- θ -graph has this property. As noted in the Introduction, there are pointsets for which no plane graph is angle-monotone with width 90 ◦ . It is an openquestion to narrow this gap and find the minimum angle γ for which every pointset has a plane graph that is angle-monotone with width γ (and thus spanningratio 1 / cos γ ).We first define the half- θ -graph. For each point u ∈ P , partition the planeinto 60 ◦ cones with apex u , with each cone defined by two rays at consecutivemultiples of 60 ◦ from the positive x -axis. Label the cones C , C , C , C , C , and C in clockwise order around u , starting from the cone containing the positive y -axis. See Figure 2(a).For two vertices u and v the canonical triangle T uv is the triangle boundedby: the cone of u that contains v ; and the line through v perpendicular to thebisector of that cone. See Figure 2(b). Notice that if v is in an even cone of u , then u is in an odd cone of v . We build the half- θ -graph as follows. For each vertex u and each even i = 0 , ,
4, add the edge uv provided that v is in the C i cone of u and T uv is empty. We call v the C i -neighbour of u . For simplicity, we assume thatno two points lie on a line parallel to a cone boundary, guaranteeing that eachvertex connects to exactly one vertex in each even cone. Hence the graph hasat most 3 n edges in total. The half- θ -graph is a type of Delaunay triangulationwhere the empty region is an equilateral triangle in a fixed orientation as opposedto a disk [8]. It can be computed in O ( n log n ) time [18]. C C C C C u C u v (a) (b) u v (c) T uv u ‘ v ‘ x Fig. 2. (a) 6 cones originating from point u , (b) Canonical triangle T uv , (c) path σ u (solid) with its empty canonical triangles shaded, path σ v (dashed) and their commonvertex x . To prove angle-monotonicity properties of the half- θ -graph, we use an idealike the one used by Angelini [3]. His goal was to show that every abstract tri-angulation has an embedding that is monotone, i.e. angle-monotone with width180 ◦ . (The same result was obtained in [14] with a different proof.) Angelini didthis by showing that the Schnyder drawing of any triangulation is monotone,nd in fact, upon careful reading, his proof shows that any Schnyder drawing isangle-monotone with the smaller width 120 ◦ . Schnyder drawings are a specialcase of half- θ -graphs [8] so it is not surprising that Angelini’s proof idea extendsto the half- θ -graph in general. Theorem 1.
The half- θ -graph is angle-monotone with width ◦ .Proof. We must prove that for any points u and v , there is an angle-monotonepath from u to v of width 120 ◦ . Assume without loss of generality that v is inthe C cone of u . See Figure 2(b).Our path from u to v will be the union of two paths, each of which is angle-monotone of width 60 ◦ . We begin by constructing a path σ u from u in whicheach vertex is joined to its C neighbour. This is a β -monotone path of width60 ◦ for β = 90 ◦ . If the path contains v we are done, so assume otherwise. Let u (cid:48) be the last vertex of the path that lies in T uv . Note that v cannot lie in the C cone of u (cid:48) . Let S be the subpath of σ u from u to u (cid:48) , together with the C coneof u (cid:48) . Then S separates T uv into two parts. Suppose that v lies in the right-handpart (the other case is symmetric). See Figure 2(c).Next, construct a path σ v from v in which each vertex is joined to its C neighbour. This is a β -monotone path of width 60 ◦ for β = 210 ◦ .We now claim that σ u and σ v have a common vertex x . Then as our finalpath from u to v we take the portion of σ u from u to x followed by the portionof σ v backwards from x to v . Since the reverse of σ v is β -monotone with width60 ◦ for β = 30 ◦ , the final path is β -monotone with width 120 ◦ for β = 60 ◦ .It remains to prove that x exists. Let v (cid:48) be the last vertex of σ v that liesstrictly to the right of S . Let u (cid:48)(cid:48) be the last vertex of σ u that lies below v (cid:48) . Weclaim that u (cid:48)(cid:48) is the C neighbour of v (cid:48) , and thus that u (cid:48)(cid:48) provides our vertex x . Let T be the empty canonical triangle from u (cid:48)(cid:48) to its C -neighbour (or theempty C cone of u (cid:48)(cid:48) in case u (cid:48)(cid:48) has no C -neighbour). First note that u (cid:48)(cid:48) is inthe C cone of v (cid:48) —otherwise v (cid:48) would be in T . Next note that T v (cid:48) u (cid:48)(cid:48) is empty—otherwise v would have a C -neighbour that is in T or is to the right of S . (cid:117)(cid:116) Theorem 1 implies that the spanning ratio of the half- θ -graph is 2, whichwas already known [11]. The best routing ratio achievable for the half- θ -graph is5 / √ ≈ .
887 [9]. (This was the first proved separation between spanning ratioand routing ratio.) Since angle-monotone paths of width 120 ◦ have spanningratio 2, this implies that no local routing algorithm can compute angle-monotonepaths with width 120 ◦ on the half- θ -graph. In this section we give a simple local “angle” routing algorithm that finds a pathfrom s to t in any triangulation. Like previous algorithms, the path walks onlyalong edges of triangles that intersect the line segment st . The novelty is thatthe next edge of the path is chosen based on angles relative to the vector st .he details of the algorithm are in Section 4.1. In Section 4.2 we provethat the algorithm has routing ratio 1 + √ Our algorithm is simple to describe: Suppose we want a route from s to t ina triangulation. Orient st horizontally, t to the right. Suppose we have reachedvertex p . Consider the last (rightmost) triangle that is incident to p and intersectsthe line segment st . The triangle has two edges incident to p . Of these two edges,take the one that has the minimum angle to the horizontal ray from p to theright. See Figure 3. Pseudo-code can be found below in Algorithm 1. Note that inthe pseudo-code, the angle test is equivalently replaced by two tests, identifyingsteps of type A and B for easier case analysis. For an example of a path computedby the algorithm, see Figure 4. Observe that the algorithm always succeeds infinding a route from s to t because it always advances rightward in the sequenceof triangles that intersect line segment st . s tp ab s tp ab rr Fig. 3.
Local routing from s to t . At vertex p , with pab being the rightmost triangleincident to p that intersects line segment st , we route from p to a because the (unsigned)angle apr is less than angle bpr . A step of type A is shown on the left and a step oftype B on the right. In this section we will prove that the above algorithm has routing ratio exactly1 + √ ◦ . Inthe last part of the section we generalize the analysis to triangulations with alarger maximum angle, and we show that the routing ratio is at least 5.07 onDelaunay triangulations.The intuition for bounding the routing ratio on Gabriel triangulations is toreplace each segment of the route by the most extreme segment possible. SeeFigure 4. Any step of type B is replaced by a 45 ◦ segment plus a horizontalsegment. Any step of type A is replaced by a vertical segment plus a horizontalsegment. Vertical segments are the bad ones, but each vertical must be preceded lgorithm 1: Local angle routing p ← s while p (cid:54) = t do Let T = pab be the rightmost triangle containing p that intersects segment st , with p and a on the same side of line st . if a is closer to line st than p then /* step of type A */ p ← a else /* step of type B */ if | slope ( pa ) | ≤ | slope ( pb ) | then p ← a else p ← b s tp q p p p q q p q q Fig. 4.
Example of route computed by Algorithm 1 (heavy blue path). In dotted red,a longer route obtained by replacing each segment of the route by the most extremeangle. Both routes are within (1 + √
2) of || st || . by 45 ◦ segments, which means that instead of travelling 1 unit horizontally (theoptimum route) we have travelled √ ◦ segment plus 1 vertically,giving us the 1 + √ e = ( p i , p i +1 ) of the path, let d x ( e ) = || x ( p i ) − x ( q p +1 ) || and d y ( e ) = || y ( p i ) − y ( p i +1 ) || . Let A (resp. B ) be the set of edges of the path wherethe algorithm makes a step of type A (resp. type B ). (Context will distinguishedge sets from steps.) Let x B = (cid:80) e ∈ B d x ( e ) and x A = (cid:80) e ∈ A d x ( e ). Lemma 3.
On any Gabriel triangulation the path computed by Algorithm 1 is x -increasing.Proof. Let us show that each step is x -increasing. Consider a step from p , with a and b as defined in Algorithm 1. Assume without loss of generality that p and a are above line st and b is below. Since T is the last triangle incident to p thatintersects st , the clockwise ordering of T is pab . Refer to Figure 3.f the algorithm takes a step of type B then a is above p (in y coordinate)and b is below p . Since ∠ bpa ≤ ◦ , thus x ( a ) and x ( b ) are greater than x ( p ). Ifthe algorithm takes a step of type A then since b is below st and a is above st and ∠ bap ≤ ◦ , thus x ( a ) is greater than x ( p ). (cid:117)(cid:116) Theorem 2.
On any Gabriel triangulation, Algorithm 1 has a routing ratio of √ and this bound is tight.Proof. We first bound (cid:80) e ∈ B || e || . Observe that each edge in B forms an anglewith the horizontal line through p that is at most 45 ◦ . Thus (cid:80) e ∈ B d y ( e ) ≤ x B and (cid:80) e ∈ B || e || ≤ √ x B .We next bound (cid:80) e ∈ A || e || . Observe that edges in A move us closer to the line st , and must be balanced by previous steps (of type B ) that moved us fartherfrom the line st . This implies that (cid:80) e ∈ A d y ( e ) ≤ (cid:80) e ∈ B d y ( e ) ≤ x B (where thelast step comes from the first observation). Since || e || ≤ d x ( e ) + d y ( e ), thus (cid:80) e ∈ A || e || ≤ x A + (cid:80) e ∈ A d y ( e ) ≤ x A + x B .Putting these together, the length of the path is bounded by (cid:80) e ∈ A || e || + (cid:80) e ∈ B || e || ≤ x A + x B + √ x B ≤ (1 + √ x A + x B ). Finally, by Lemma 3, x A + x B = || st || , so this proves that the routing ratio is at most (1 + √ (cid:117)(cid:116) We conclude this section with two results on the behaviour of the routingalgorithm on other triangulations. Proofs are deferred to Appendix B.
Theorem 3.
In a triangulation with maximum angle α < ◦ Algorithm 1 hasa routing ratio of (sin α + sin α ) / sin α and this bound is tight. Theorem 4.
The routing ratio of Algorithm 1 on Delaunay triangulation isgreater than . . We believe that the routing ratio of Algorithm 1 on Delaunay triangulationsis close to 5 .
07, but leave that as an open question. We remark that Algorithm 1is different from the generalization of Chew’s Routing Algorithm for Delaunaytriangulations [10] (cf. the algorithm described in [7]).
In this section we prove some limits on local routing on Gabriel triangulations.Proofs are deferred to Appendix B.A routing algorithm on a geometric graph G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to anyvertex t is at most c times the length of the shortest path from s to t in G , and c is the minimum such value. (Recall that the routing ratio compares the lengthof the path produced by the algorithm to the Euclidean distance between theendpoints. Thus the competitive ratio is less than or equal to the routing ratio.)A routing algorithm is k -local (for some integer constant k >
0) if it makesforwarding decisions based on: (1) the k -neighborhood in G of the current posi-tion of the message; and (2) limited information stored in the message header. heorem 5. Any k -local routing algorithm on Gabriel triangulations has rout-ing ratio at least 1.4966 and competitive ratio at least 1.2687. Although Gabriel triangulations are angle-monotone [12], Theorem 5 showsthat no local routing algorithm can compute angle-monotone paths since thatwould give routing ratio √
2. The following theorem tells us that even less con-strained paths cannot be computed locally:
Theorem 6.
There is no k -local routing algorithm on Gabriel triangulationsthat always finds self-approaching paths. We conclude this paper with some open questions.1. What is the minimum angle γ for which every point set has a plane geometricgraph that is angle-monotone with width γ (and thus has spanning ratio1 / cos γ )? We proved γ ≤ ◦ , and it is known that γ > ◦ .2. Is there a local routing algorithm with bounded routing ratio for any angle-monotone graph? Any increasing-chord graph?3. We bounded the routing ratio of our local routing algorithm on triangula-tions based on the maximum angle in the triangulation, but how does thisrelate to the property of being generalized angle-monotone? If a triangula-tion has bounded maximum angle, is it generalized angle-monotone? Theonly thing known is that maximum angle 90 ◦ implies angle-monotone withwidth 90 ◦ [12].4. Is the standard Delaunay triangulation generalized angle-monotone? In par-ticular, proving that the Delaunay triangulation is angle-monotone withwidth strictly less than 120 ◦ would provide a different proof that the Delau-nay triangulation has spanning ratio less than 2 [21]. It is known that theDelaunay triangulation is not angle-monotone with width 90 ◦ (see Section 1).5. How does our local routing algorithm behave on standard Delaunay trian-gulations? We proved a lower bound of 5.07 on the routing ratio. We believethe routing ratio is close to this value, but have no upper bound. Acknowledgements
This work was begun at the CMO-BIRS Workshop on Searching and Routingin Discrete and Continuous Domains, October 11–16, 2015. We thank the otherparticipants of the workshop for many good ideas and stimulating discussions.We thank an anonymous referee for helpful comments.Funding acknowledgements: A.L. thanks NSERC (Natural Sciences and En-gineering Council of Canada). S.V. thanks NSERC and the Ontario Ministry ofResearch and Innovation. N.B. thanks French National Research Agency (ANR)in the frame of the “Investments for the future” Programme IdEx Bordeaux- CPU (ANR-10-IDEX-03-02). I.K. was supported in part by the NWO underproject no. 612.001.106, and by F.R.S.-FNRS. eferences
1. Aichholzer, O., Aurenhammer, F., Icking, C., Klein, R., Langetepe, E., Rote, G.:Generalized self-approaching curves. Discrete Applied Mathematics 109(1–2), 3–24(2001)2. Alamdari, S., Chan, T.M., Grant, E., Lubiw, A., Pathak, V.: Self-approachinggraphs. In: Didimo, W., Patrignani, M. (eds.) Proc. Graph Drawing (GD), LNCS,vol. 7704, pp. 260–271. Springer (2013)3. Angelini, P.: Monotone drawings of graphs with few directions. In: 6th Int. Conf.Information, Intelligence, Systems and Applications (IISA). pp. 1–6. IEEE (2015)4. Angelini, P., Colasante, E., Battista, G.D., Frati, F., Patrignani, M.: Monotonedrawings of graphs. J. Graph Algorithms Appl. 16(1), 5–35 (2012)5. Angelini, P., Frati, F., Grilli, L.: An algorithm to construct greedy drawings oftriangulations. J. Graph Algorithms Appl. 14(1), 19–51 (2010)6. Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. In: Proc. 31stSymp. on Foundations of Computer Science (FOCS). pp. 231–241. IEEE (1990)7. Bonichon, N., Bose, P., De Carufel, J.L., Perkovi´c, L., Van Renssen, A.: Upperand lower bounds for online routing on Delaunay triangulations. In: Bansal, N.,Finocchi, I. (eds.) Proc. 23rd European Symp. on Algorithms (ESA). LNCS, vol.9294, pp. 203–214. Springer (2015)8. Bonichon, N., Gavoille, C., Hanusse, N., Ilcinkas, D.: Connections between theta-graphs, Delaunay triangulations, and orthogonal surfaces. In: Thilikos, D.M. (ed.)Proc. 36th Int. Workshop Graph Theoretic Concepts in Computer Science (WG).LNCS, vol. 6410, pp. 266–278 (2010)9. Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Optimal local routing onDelaunay triangulations defined by empty equilateral triangles. SIAM J. Comput.44(6), 1626–1649 (2015)10. Chew, L.P.: There is a planar graph almost as good as the complete graph. In:Proc. 2nd Annual Symp. Computational Geometry (SoCG). pp. 169–177 (1986)11. Chew, L.P.: There are planar graphs almost as good as the complete graph. J.Computer and System Sciences 39(2), 205–219 (1989)12. Dehkordi, H.R., Frati, F., Gudmundsson, J.: Increasing-chord graphs on point sets.J. Graph Algorithms Appl. 19(2), 761–778 (2015)13. Dumitrescu, A., Ghosh, A.: Lower bounds on the dilation of plane spanners (2015),http://arxiv.org/pdf/1509.07181v3.pdf14. He, X., He, D.: Monotone drawings of 3-connected plane graphs. In: Bansal, N.,Finocchi, I. (eds.) Proc. 23rd European Symp. on Algorithms (ESA). LNCS, vol.9294, pp. 729–741. Springer (2015)15. Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Cam-bridge Philosophical Society 125, 441–453 (1995)16. Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces.Discrete Comput. Geom. 44, 686–705 (2010)17. Mulzer, W.: Minimum Dilation Triangulations for the Regular n-Gon. Master’sthesis, Freie Universit¨at Berlin (2004)18. Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge UniversityPress (2007)19. Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing.Theor. Comput. Sci. 344, 3–14 (2005)20. Rote, G.: Curves with increasing chords. Math. Proc. Cambridge PhilosophicalSociety 115, 1–12 (1994)1. Xia, G.: The stretch factor of the Delaunay triangulation is less than 1.998. SIAMJ. Comput. 42(4), 1620–1659 (2013)
A Omitted Proofs for Section 1
Proof (Proof of Observation 1).
In the worst case we travel the two equal sidesof an isoceles triangle with base length 1 and two angles of γ/
2. If (cid:96) is the sidelength, the ratio is 2 (cid:96) , and we have cos γ = /(cid:96) . Thus the ratio is 1 / cos γ . (cid:117)(cid:116) u vβ Fig. 5.
Illustration of Lemma 1. A β -monotone path (in blue) inside the rectangle withboth sides at angles β ± ◦ . This rectangle lies inside the disc of diameter uv . B Omitted Proofs for Section 4
Proof (Proof of Theorem 2).
To complete the proof, we show an example forwhich our algorithm gives a routing ratio of 1 + √
2. Consider the configurationshown in Figure 6. It is a Gabriel triangulation and the route computed by thealgorithm is as shown. Observe that the size of the leftmost circle can be madearbitrarily small compared to || st || . Hence, when s = (0 ,
0) and t = (1 , s → (1 , → t . Thus we can builda point set such that the length of the computed route is as close to 1 + √ (cid:117)(cid:116) Proof (Proof sketch for Theorem 3).
Following the intuitive justification for therouting ratio of Algorithm 1 on Gabriel triangulations, lengthen the route byreplacing each segment of the route by the most extreme segment possible. Anystep of type B is replaced by a segment at angle α plus a horizontal segment. Anystep of type A is replaced by a segment at angle α plus a horizontal segment.In all cases angles are measured from the forward horizontal. See Figure 7.Segments of type A are the bad ones, but each such segment must be preceded tq p q p Fig. 6.
Example that gives a lower bound on the routing ratio of our routing algorithmon Gabriel triangulations. The route found by the algorithm is drawn as a heavy bluepath. by angle α segments, which means that instead of travelling 1 unit horizontally(the optimum route) we have travelled on a segment of angle α and then ona segment of angle − α (both angles measured w.r.t the forward horizontal).Let these segments have lengths (cid:96) and (cid:96) respectively. In the triangle formedby these three segments, the (cid:96) segment is opposite angle α , the (cid:96) segmentis opposite angle α and the unit horizontal is opposite angle 180 ◦ − α . Weneed 180 ◦ − α >
0, i.e. α < ◦ . By the sine law, (cid:96) = sin α/ sin α and (cid:96) = sin α / sin α . Thus the distance travelled is (cid:96) + (cid:96) = (sin α + sin α ) / sin α .To show that the bound is tight, we generalize the example of Figure 6. Theresulting example is shown in Figure 8. (cid:117)(cid:116) s tp q p p p q q p q q Fig. 7.
Intuition for general routing. s tq p q p Fig. 8.
The worst case situation for general routing.
Proof (Proof of Theorem 4).
Let us explain the example of Figure 9. This De-launay triangulation is defined in the following way: The first triangle sp q issuch that the slope of line sp is slightly smaller than the slope of sq , so weroute to p . Let q be a point on the empty circle C containing s , p and q that is slightly below the x -axis. Let C be the circle that goes through p and q such that the tangent of C at q is horizontal. Let p be a point on C suchthat the slope of p p is slightly smaller than the slope of p q . We place point t at the rightmost intersection of C and the x -axis, and we place vertices denselyon the arc of C between p and t . The route in the example of Figure 9 haslength about 5 || st || . Moving p closer and closer to s leads to 5 .
07 as a lowerbound on the routing ratio of Algorithm 1 on Delaunay triangulations. (cid:117)(cid:116)
Proof (Proof of Theorem 5).
Let us consider the triangulation of Figure 10. This triangulation is definedas follows: all the triangles intersecting the segment st are right triangles. Thefirst one is isosceles and symmetric with respect to the x -axis. Then we have a fan of 2 k − q and q (cid:48) be respectively the upper rightmost andlower rightmost points of this set of triangles. The next triangle qq (cid:48) B is suchthat the angle ∠ q (cid:48) qB = 22 . ◦ . The point t is on the intersection of the line q (cid:48) B and the x -axis. We complete the triangulation with two triangles, qBA and ABt having common hypotenuse AB . Finally we make the fan of of 2 k − || qq (cid:48) || = 2.Now let us consider any deterministic k -local routing algorithm. We considertwo triangulations: The first is the one described above (and shown in Fig. 10)and the second one is obtained from the first by reflecting over the x -axis thepart of the triangulation that lies to the right of qq (cid:48) . No deterministic k -localrouting algorithm computing a path from s to t can distinguish between the two tq p q p Fig. 9.
Example that gives a 5.0 lower bound on the routing ratio of Algorithm 1 onDelaunay triangulations. point sets until a vertex less than k hops away from q or q (cid:48) is reached. Let q (cid:48)(cid:48) bethe vertex k hops away from q or q (cid:48) that is reached by the algorithm on eithertriangulation.Since the fan is arbitrarily thin, q (cid:48)(cid:48) can be assumed to be arbitrarily close to q or to q (cid:48) .Each case, q or q (cid:48) , leads to a non-optimal path for one of the point sets;we only consider the first case as the second will follow by symmetry. If q (cid:48)(cid:48) isarbitrarily close to q then, for the point set shown in Fig. 10, the shortest pathsfrom q (cid:48)(cid:48) to t go through A or B and are of length || qB || + || Bt || = 2 || qB || =4 cos(22 . ◦ ). Moreover || sq || = √ || st || = 1+1 / tan(22 . ◦ ). Hence the lengthof the complete path computed by the algorithm is at least √ . ◦ )1+1 / tan(22 . ◦ ) || st || ≈ . s to t goes through q (cid:48) and is of length || sq (cid:48) || + || q (cid:48) B || + || q (cid:48) t || = √ − cos(45 ◦ )) / sin(22 . ◦ ) + 2 cos(22 . ◦ ). Thus a lower bound on the competitive ratiois √ . ◦ ) √ − cos(45 ◦ )) / sin(22 . ◦ )+2 cos(22 . ◦ ) ≈ . (cid:117)(cid:116) Proof (Proof sketch for Theorem 6).
We apply reasoning as in the previous proof,but this time on the triangulation of Figure 11, where the fat segment qq (cid:48) rep-resents a fan of 2 k − q (if not we consider the symmetric triangulation with respectto the x -axis). Moving from s to q the distance toward B is not decreasing.Hence a self approaching path that goes through the edge sq cannot go throughthe vertex B . Hence once at vertex q the only possibility is to use the edge qA .But moving along the edge qA the distance toward t is not decreasing. Hence,there is no self approaching path from s to t that goes through q .So for any deterministic k -local routing algorithm, there exists a triangulationon which the algorithm will not find a self-approaching path. (cid:117)(cid:116) . ◦ q q (cid:48) B A ts
Fig. 10.
Example for lower bounds on the routing ratio and competitive ratio of any k -local routing algorithm on Gabriel triangulations. q q (cid:48) B A ts
Fig. 11.
Example of Gabriel triangulation used to show that no kk