Stable Delaunay Grahs Pankaj K. Agarwal Jie Gao Leonidas Guibas Haim Kalan Natan Rubin Micha Sharir Aril 29, 2014 Abstract Let P be a set of n oints in R 2, and let DT(P) denote its Euclidean Delaunay triangulation. We introduce the notion of an edge of DT(P) being stable. Defined in terms of a arameter α > 0, a Delaunay edge is called α-stable, if the (eual) angles at which and see the corresonding Voronoi edge e are at least α. A subgrah G of DT(P) is called (cα, α)-stable Delaunay grah (SDG in short), for some constant c 1, if every edge in G is α-stable and every cα-stable of DT(P) is in G. We show that if an edge is stable in the Euclidean Delaunay triangulation of P, then it is also a stable edge, though for a different value of α, in the Delaunay triangulation of P under any convex distance function that is sufficiently close to the Euclidean norm, and vice-versa. In articular, a 6α-stable edge in DT(P) is α-stable in the Delaunay triangulation under the distance function induced by a regular k-gon for k 2π/α, and vice-versa. Exloiting this relationshi and the analysis in [3], we resent a linear-size kinetic data structure (KDS) for maintaining an (8α, α)-sdg as the oints of P move. If the oints move along algebraic trajectories of bounded degree, the KDS rocesses nearly uadratic events during the motion, each of which can rocessed in O(log n) time. Finally, we show that a number of useful roerties of DT(P) are retained by SDG of P. An earlier version of this aer aeared in Proc. 26th Annual Symosium on Comutational Geometry, 2010, 127 136. Deartment of Comuter Science, Duke University, Durham, NC 27708-0129, USA, ankaj@cs.duke.edu. Deartment of Comuter Science, Stony Brook University, Stony Brook, NY 11794, USA, jgao@cs.sunysb.edu. Deartment of Comuter Science, Stanford University, Stanford, CA 94305, USA, guibas@cs.stanford.edu. School of Comuter Science, Tel Aviv University, Tel Aviv 69978, Israel. haimk@tau.ac.il. Jussieu Institute of Mathematics, Pierre and Marie Curie University and Paris Diderot University, UMR 7586 du CNRS, Paris 75005, France. rubinnat.ac@gmail.com. School of Comuter Science, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. michas@tau.ac.il. 1
1 Introduction Let P be a set of n oints in R 2. For a oint P, the (Euclidean) Voronoi cell of is defined as Vor() = {x R 2 x x P}. The Voronoi cells of oints in P are nonemty, have airwise-disjoint interiors, and artition the lane. The lanar subdivision induced by these Voronoi cells is referred to as the (Euclidean) Voronoi diagram of P and we denote it as VD(P). The Delaunay grah of P is the dual grah of VD(P), i.e., is an edge of the Delaunay grah if and only if Vor() and Vor() share an edge. This is euivalent to the existence of a circle assing through and that does not contain any other oint of P in its interior any circle centered at a oint of Vor() Vor() and assing through and is such a circle. If no four oints of P are cocircular, then the lanar subdivision induced by the Delaunay grah is a triangulation of the convex hull of P the well-known (Euclidean) Delaunay triangulation of P, denoted as DT(P). See Figure 1 (a). DT(P) consists of all triangles whose circumcircles do not contain oints of P in their interior. Delaunay triangulations and Voronoi diagrams are fundamental to much of comutational geometry and its alications. See [8] for a very recent textbook on these structures. (a) (b) Figure 1. (a) The Euclidean Voronoi diagram (dotted) and Delaunay triangulation (solid); (b) the Q-Voronoi diagram and Q-Delaunay triangulation for an axis-arallel suare Q, i.e., the Voronoi and Delaunay diagrams under the L -metric. In many alications of Delaunay/Voronoi methods (e.g., mesh generation and kinetic collision detection), the inut oints are moving continuously, so they need to be efficiently udated as motion occurs. Even though the motion of the oints is continuous, the combinatorial and toological structure of VD(P) and DT(P) change only at discrete times when certain critical events occur. A challenging oen uestion in combinatorial geometry is to bound the number of critical events if each oint of P moves along an algebraic trajectory of constant degree. Guibas et al. [14] showed a roughly cubic uer bound of O(n 2 λ s (n)) on the number of critical events. Here λ s (n) is the maximum length of an (n, s)-davenort-schinzel seuence [21], and s is a constant deending on the degree of the motion of the oints. See also Fu and Lee [13]. The best known lower bound is uadratic [21]. Recent works of Rubin [19, 20] establish an almost uadratic bound of O(n 2+ε ), for any ε > 0, for the restricted cases where any four oints of P can be cocircular at most two or three times. In articular, the latter study [20] covers the case of oints moving along lines at common unit seed, which has been highlighted as a major oen roblem in discrete and comutational geometry; see [12]. Nevertheless, no sub-cubic uer bound is known for more 1
general motions, including the case where the oints of P are moving along lines at non-uniform, albeit fixed, seeds. It is worth mentioning that the analysis in [19], and even more so in [20], is fairly involved, which results in a huge imlicit constant of roortionality. Given this ga in the bound on the number of critical events, it is natural to ask whether one can define a large subgrah of the Delaunay grah of P so that (i) it rovably exeriences at most a nearly uadratic number of critical events, (ii) it is reasonably easy to define and maintain, and (iii) it retains useful roerties for further alications. This aer defines such a subgrah of the Delaunay grah, shows that it can be maintained efficiently, and roves that it reserves a number of useful roerties of DT(P). Related work. It is well known that DT(P) can be maintained efficiently using the so-called kinetic data structure framework roosed by Basch et al. [10]. A triangulation T of the convex hull conv(p) of P is the Delaunay triangulation of P if and only if for every edge adjacent to two triangles r + and r in T, the circumcircle of r + (res., r ) does not contain r (res., r + ). Euivalently, r + + r < π. (1) Euality occurs when,, r +, r are cocircular, which generally signifies that a combinatorial change in DT(P) (a so-called edge fli) is about to take lace. We also extend (1) to aly to edges of the hull, each having only one adjacent triangle, r +. In this case we take r to lie at infinity, and ut r = 0. An euality in (1) occurs when,, r + become collinear (along the hull boundary), and again this signifies a combinatorial change in DT(P). This makes the maintenance of DT(P) under oint motion uite simle: an udate is necessary only when the emty circumcircle condition (1) fails for one of the edges, i.e., for an edge, adjacent to triangles r + and r,,, r +, and r become cocircular. 1 Whenever such an event haens, the edge is flied with r + r to restore Delaunayhood. Keeing track of these cocircularity events is straightforward, and each such event is detected and rocessed in O(log n) time [14]. However, as mentioned above, the best known uer bound on the number of events rocessed by this KDS (which is the number of toological changes in DT(P) during the motion), assuming that the oints of P are moving along algebraic trajectories of bounded degree, is near cubic [14] (excet for the secial cases treated in [19, 20]). So far we have only considered the Euclidean Voronoi and Delaunay diagrams, but a considerable amount of literature exists on Voronoi and Delaunay diagrams under other norms and so-called convex distance functions; see Section 2 for details. Chew [11] showed that the Delaunay triangulation of P under the L 1 - or L -metric exeriences only a near-uadratic number of events, if the motion of the oints of P is algebraic of bounded degree. In the comanion aer [3], we resent a kinetic data structure for maintaining the Voronoi diagram and Delaunay triangulation of P under a olygonal convex distance function for an arbitrary convex olygon Q. (see Section 2 for the definition) that rocesses only a near-uadratic number of events, and can be udated in O(log n) time at each event. Since a regular convex k-gon aroximates a circular disk, it is temting to maintain the Delaunay triangulation under a olygonal convex distance function as a (hoefully substantial) ortion of the Euclidean Delaunay grah of P. Unfortunately, the former is not necessarily a subgrah of the latter [8]. 1 We assume the motion of the oints to be sufficiently generic, so that no more than four oints can become cocircular at any given time, and so that euality in (1) is not a local maximum of the left-hand side. 2
Many subgrahs of DT(P), such as the Euclidean minimum sanning tree (MST), Gabriel grah, relative neighborhood grah, and α-shaes, have been used extensively in a wide range of alications (see e.g. [8]). However no sub-cubic bound is known on the number of discrete changes in their structures under an algebraic motion of the oints of P of bounded degree. Furthermore, no efficient kinetic data structures are known for maintaining them, for unlike DT(P), they may undergo a non-local change at a critical event; see [1, 9, 18] for some artial results on maintaining an MST. Our results. Stable Delaunay edges: We introduce the notion of α-stable Delaunay edges, for a fixed arameter α > 0, defined as follows. Let be a Delaunay edge under the Euclidean norm, and let r + and r be the two Delaunay triangles incident to. Then is called α-stable if its oosite angles in these triangles satisfy r + + r π α. (2) As above, the case where lies on conv(p) is treated as if r, say, lies at infinity, so that the corresonding angle r is eual to 0. An euivalent and more useful definition, in terms of the dual Voronoi diagram, is that is α-stable if the angles at which and see their common Voronoi edge e are at least α each. See Figure 2(a). In the case where lies on conv(p) the corresonding dual Voronoi edge e is an infinite ray emanating from some Voronoi vertex x. We define the angle in which a oint see such a Voronoi ray to be the angle between the segment x and an infinite ray arallel to e emanating from. With this definition of the angle in which a oint of conv(p) sees a Voronoi ray, it is easy to check that the alternative definition of α-stability is euivalent to the ordinal definition also when lies on conv(p). We call the Voronoi edges corresonding to α-stable Delaunay edges α-long (and call the remaining edges α-short). See Figure 2. Note that for α = 0, when no four oints are cocircular, (2) coincides with (1). α r α a r + D b α D + (a) (b) Figure 2. (a) An α-stable edge in DT(P), and its dual edge e = ab in VD(P); the fact that r + + r = π α follows by elementary geometric considerations. (b) A SDG of the Delaunay triangulation DT(P) shown in Figure 1 (a), with α = π/8. A justification for calling such edges stable lies in the following observation: If a Delaunay edge is α-stable then it remains in DT(P) during any continuous motion of the oints of P for which every angle r, for r P \ {, }, changes by at most α/2. This is clear because, as is easily verified, at any time when is α-stable we have r + + r π α for any air of oints r +, 3
r lying on oosite sides of the line l suorting, so, if each of these angles changes by at most α/2 we still have r + + r π for every such air r +, r, imlying that remains an edge of DT(P). 2 Hence, as long as the small angle change condition holds, stable Delaunay edges remain a long time in the triangulation. Informally seaking, the non-stable edges of DT(P) are those for which and are almost cocircular with their two common Delaunay neighbors r +, r, and hence is more likely to get flied soon. Stable Delaunay grah: For two arameters 0 α α π, we call a subgrah G of DT(P) an (α, α)-stable Delaunay grah (an (α, α)-sdg for short) if (S1) every edge of G is α-stable, and (S2) every α -stable edge of DT(P) belongs to G. Note that an (α, α)-sdg is not uniuely defined even for fixed α, α because the edges that are α-stable but not α -stable may or may not be in G. Throughout this aer, α will be some fixed (and reasonably small) multile of α. 1 D I cos α o D O Q Figure 3. A convex distance function, induced by Q, that is α-close to the Euclidean norm. Our main result is that a stable edge of the Euclidean Delaunay triangulation aears as stable edge in the Delaunay triangulation under any convex distance function that is sufficiently close to the Euclidean norm (see Section 2 for more details). More recisely, we say that the distance function induced by a comact convex set Q is α-close to the Euclidean norm if Q is contained in the unit disk D O and contains the disk D I = (cos α)d O both centered in the origin. 3 See Figure 3. In articular, for k = π/α, the regular k-gon Q k is such a set, as easy trigonometry shows. We rove the following: Theorem 1.1. Let P be a set of n oints in R 2, α > 0 a arameter, and Q a comact, convex set inducing a convex distance function d Q (, ) that is α-close to the Euclidean norm. Then the following roerties hold. (i) Every 11α-stable Delaunay edge under the Euclidean norm is an α-stable Delaunay edge under d Q. (ii) Symmetrically, every 11α-stable Delaunay edge under d Q is also an α-stable Delaunay edge under the Euclidean norm. 2 This argument also covers the cases when a oint r crosses l from side to side: Since each oint, on either side of l, sees at an angle of π α, it follows that no oint can cross itself the angle has to increase from π α to π. Any other crossing of l by a oint r causes r to decrease to 0, and even if it increases to α/2 on the other side of l, is still an edge of DT, as is easily checked. 3 The Hausdorff distance between Q and D O is at most 1 cos α α 2 /2. 4
In articular, if Q is a regular k-gon for k 2π/α, then the above theorem holds for Q. In the comanion aer [3], we have resented an efficient kinetic data structure for maintaining the Delaunay triangulation and Voronoi diagram of P under a olygonal convex distance function. Using this result, we obtain the second main result of the aer: Theorem 1.2. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, and let α (0, π) be a arameter. A Euclidean (8α, α)-stable Delaunay grah of P can be maintained by a linear-size KDS that rocesses O ( 1 nλ α 4 r (n) ) events and udates the SDG at each event in O(log n) time. Here r is a constant that deends on the degree of the motion of the oints of P, and λ r (n) is the maximum length of a Davenort-Schnizel seuence of order r. For simlicity, we first rove in Section 3 Theorem 1.1 for the case when Q is a regular k-gon, for k 2π/α, and use the argument to rove Theorem 1.2. Actually, we rove Theorem 1.1 with a slightly better constant using the additional structure ossessed by the diagrams when Q is a regular k-gon. Next, we rove in Section 4 Theorem 1.1 for an arbitrary Q. Finally, we rove in Section 5 a few useful roerties of DT(P) that are retained by the stable Delaunay grah of P. 2 Preliminaries This section introduces a few notations, definitions, and known results that we will need in the aer. We reresent a direction in R 2 as a oint on the unit circle S 1. For a direction u S 1 and an angle θ [0, 2π), we use u + θ to denote the direction obtained after rotating the vector ou by angle θ in clockwise direction. For a oint x R 2 and a direction u S 1, let u[x] denote the ray emanating from x in direction u. Q-distance function. Let Q be a comact, convex set with non-emty interior and with the origin, denoted by o, lying in its interior. A homothetic coy Q of Q can be reresented by a air (, λ), with the interretation Q = + λq; is the lacement (location) of the center o of Q, and λ is its scaling factor (about its center). Q defines a distance function (also called the gauge of Q) d Q (x, y) = min{λ y x + λq}. Note that, unless Q is centrally symmetric with resect to the origin, d Q is not symmetric. Given a finite oint set P R 2 and a oint P, we denote by Vor Q (), VD Q (P), and DT Q (P) the Voronoi cell of, the Voronoi diagram of P, and the Delaunay triangulation of P, resectively, under the distance function d Q (, ); see Figure 1 (b). To be recise (because of the otential asymmetry of d Q ), we define Vor() = {x R 2 d Q (x, ) d Q (x, ) P}, and then VD Q (P) and DT Q (P) are defined in comlete analogy to the Euclidean case. We refer the reader to [3] for formal definitions and details of these structures. Throughout this aer, we will dro the suerscrit Q from Vor Q, VD Q, DT Q when referring to them under the Euclidean norm. For a oint z R 2, let Q[z] denote the homothetic coy of Q centered at z such that its boundary touches the Q-nearest neighbor(s) of z in P, i.e., Q[z] is reresented by the air (z, λ) 5
Q (u) b Q Q,c (u) c u[] c u[] r b Q r c u [] Q r (u ) l (u) Q r (u) b Q Figure 4. (a) Homothetic coies Q (u), centered at c u[] (dashed), and Q (u) (solid) of Q. The ray u[] hits b Q at the center c of Q (u), which is defined, when Q is smooth at its contact with, if and only if and Q (u) lie on the same side of l (u); (b) Sliding Q r (u) away from. where λ = min P d Q (z, ). In other words, Q[z] is the largest homothetic coy of Q that is centered at z whose interior is P-emty. We also use the notation Q (u) to denote a generic homothetic coy of Q which touches and is centered at some oint on u[]. See Figure 4 (a). Note that all homothetic coies of Q (u) touch at the same oint ζ of Q, and therefore share the same tangent at. This tangent is uniue if Q is smooth at ζ, and we denote it by l (u). If Q is not smooth at ζ then there is a nontrivial range of tangents (i.e., suorting lines) to Q at ζ; we can take l (u) to be any of them, and it will be a suorting line to all the coies Q (u) of Q. For a air of oints, P, let b Q denote the Q-bisector of and the locus of all lacements of the center of any homothetic coy Q of Q that touches and. If Q is strictly convex or if Q is not strictly convex but no two oints are collinear with a straight segment on Q, then b Q is a one-dimensional curve and any ray u[] that hits b Q does so at a uniue oint. For such a direction u and a air of oints, P, let Q (u) denote the homothetic coy of Q that touches and, whose center is u[] b Q. If Q is not strictly convex and, are oints in P such that is arallel to a straight ortion e of Q then b Q is not one-dimensional. In this case Q (u) is not well defined when u is a direction that connects e to the center c of Q. As is easy to check, in any other case the ray u[] either hits b at a uniue oint which determines Q (u), or entirely misses b Q. See the comanion aer [3] for a detailed discussion of this henomenon. A useful roerty of the Q-bisectors is that any two bisectors b Q, b Q r with a common generating oint, intersect exactly once, namely, at the center c of the uniue homothetic coy of Q that simultaneously touches, and r [17]. For this roerty to hold, though, we need to assume that (i) the oints,, r are not collinear, and (ii) either Q is strictly convex, or, otherwise, that none of the directions, r is arallel to a straight ortion of Q. (The recise condition is that b and b r be one-dimensional in a neighborhood of c.) The local toology of the restricted Q-Voronoi diagram VD Q ({,, r}) near c is largely determined by the orientation of the triangle r. Secifically, assume with no loss of generality that r is counterclockwise to, and let u be the direction of the ray c. Refer to Figure 4 (b). If we continuously rotate a ray u[], for u S 1, in counterclockwise direction from u [], the corresonding coy Q r (u) will slide away from its contact with because the ortion of Q r (u) to the right of r shrinks during the rotation. Therefore, the rotating ray u[] either misses b Q entirely or hits b Q after b Q r. A symmetric henomenon, with and r interchanged, 6
takes lace if we rotate the ray u[] in clockwise direction from u []. It is known that Vor Q (), for every oint P, is star-shaed [8], which imlies that each Voronoi edge e Q is fully enclosed between the two rays that emanate from, or from, through its endoints. We remark that, unlike the Euclidean case, the angles xy, xy need not be eual in general. Finally, we extend the notion of stable edges to DT Q (P). We call an edge DT Q (P) α-stable if the following roerty holds for the dual edge e Q in VD Q (P): Each of the oints, sees their common Q-Voronoi edge e Q at angle at least α. That is, if x and y are the endoints of e Q, then min{ xy, xy} α. This definition coincides with the definition of α-stability under the Euclidean norm when Q is the unit disk (and in this case both angles are eual). Remark. If Q is not strictly convex (and is arallel to a straight ortion of Q), the endoints of e Q may be not well defined. In this case, we resort to the following, more careful definition of α-stability. A ray u[] is said to (roerly) cross e Q only if the coy Q (u) is uniuely defined. The center of such a coy Q (u) necessarily lies within the one-dimensional ortion ẽ Q of Q (u), which is easily seen to be non-emty and connected. We say that the edge e Q is α-stable if the set of rays u[] roerly crossing e Q sans an angle of at least α, and a symmetric condition holds for the rays emanating from. In other words, our notion of α-stability ignores the two-dimensional regions of e Q (if these exist). 4 Polygonal convex distance function. As mentioned in the introduction, we will be considering the case when Q is a regular k-gon, for some even integer k 2π/α, centered at the origin. Let v 0,..., v k 1 be its seuence of vertices arranged in clockwise direction. For each 0 j < k, let u j be the direction of the vector that connects v j to the center of Q (see Figure 5 (a)). We will use b, Vor (), VD (P), DT (P) to denote b Q, Vor Q (), VD Q (P), DT Q (P), resectively, when Q is a regular k-gon. We say that P is in general osition (with resect to Q) if no three oints of P lie on a line, no two oints of P lie on a line arallel to an edge or a diagonal of Q, and no four oints of P are Q-cocircular, i.e., no four oints of P lie on the boundary of a common homothetic coy of Q. The lacements on b at which (at least) one of and, say,, touches Q at a vertex is called a corner lacement (or a corner contact) at ; see Figure 5 (b). We also refer to these oints on b as breakoints. We call a homothetic coy of Q whose vertex v j touches a oint, a v j -lacement of Q at. The following roerty of b is roved in [3, Lemma 2.5]: Lemma 2.1. Let Q be a regular k-gon, and let and be two oints in general osition with resect to Q. Then b is a olygonal chain with k 2 breakoints and the breakoints along b alternate between corner contacts at and corner contacts at. 4 As is easy to check, the one-dimensional ortion ẽ Q of e Q varies continuously (in Hausdorff sense) with any sufficiently small erturbation of and within P. Furthermore, it is the only such ortion: If a ray u[] hits e Q outside ẽ Q (i.e., within its two-dimensional ortion), there is a symbolic erturbation of and causing u[] to comletely miss e Q. 7
Q v 3 v 2 u 4 v 1 α o v 0 v 7 b v 4 v 5 v 6 (a) (b) Figure 5. (a) A regular octagon Q centered at the origin. (b) The bisector b for the regular octagon Q; it has six breakoints and the corner contacts along b alternate between contacts at (hollow circles) and contacts at (filled circles). 3 DT (P) and Euclidean SDG s In this section we first rove a slightly stronger version of Theorem 1.1 for the case when Q is a regular k-gon for some even integer k 2π/α, and then rove Theorem 1.2. We follow the notation in Section 2, and, for simlicity, we assume that α = 2π/k. ϕ j [, ] [, ] ϕ j u j [] b u j [] b u j [] (a) (b) (c) Figure 6. (a) The function ϕ j [, ], which is eual to the radius of the circle that ass through and and whose center lies on u j []. (b) The bisector b and the function ϕ j [, ], which is eual to the radius of the circle that circumscribes the v j -lacement of Q at that also touches. (c) The case when ϕ j [, ] = while ϕ j[, ] <. In this case lies in one of the shaded wedges. For any air, P, let ϕ j [, ] denote the distance from to the oint u j [] b ; we ut ϕ j [, ] = if u j [] does not intersect b. See Figure 6 (a). The oint minimizes ϕ i [, ], among all oints for which u i [] intersects b, if and only if the intersection between b and u i [] lies on the Voronoi edge e. We call the neighbor of in direction u i, and denote it by N i (). Similarly, let ϕ j [, ] denote the distance from to the oint u j[] b ; we ut ϕ j [, ] = if u j [] does not intersect b. If ϕ j [, ] < then the oint b u j [] is the center of the v j -lacement Q of Q at that also touches, and there is a uniue such oint. The value ϕ j [, ] is eual to the circumradius of Q. See Figure 6 (b). The neighbor Nj [] of in direction u j is defined to be the oint P \ {} that minimizes ϕ j [, ]. Note that ϕ j [, ] < if and only if the angle between and u j [] is smaller than π/2. In contrast, ϕ j [, ] < if and only if the angle between and u j[] is at most π/2 π/k = 8
π/2 α/2. Moreover, we have ϕ j [, ] ϕ j [, ] (see Figure 6). Therefore, ϕ j [, ] < always imlies ϕ j [, ] <, but not vice versa; see Figure 6 (c). Note also that in both the Euclidean and the olygonal cases, the resective uantities N j [] and Nj [] may be undefined. Lemma 3.1. Let, P be a air of oints such that N j () = for h 3 consecutive indices, say 0 j h 1. Then for each of these indices, excet ossibly for the first and the last one, we also have Nj [] =. Proof. Let w 1 (res., w 2 ) be the oint at which the ray u 0 [] (res., u h 1 []) hits the edge e in VD(P). (By assumtion, both oints exist.) Let D 1 and D 2 be the disks centered at w 1 and w 2, resectively, and touching and. By definition, neither of these disks contains a oint of P in its interior. The angle between the tangents to D 1 and D 2 at or at (these angles are eual) is β = (h 1)α; see Figure 7 (a). u j [] w 1 D 1 D 2 w 2 D β β D + l e 1 a1 Q j D 1 τ (a) (b) (c) e a 2 D + 2 l D 2 Figure 7. (a) The angle between the tangents to D 1 and D 2 at (or at ) is eual to w 1 w 2 = β = (h 1)α. (b) The disks D and D + and the homothetic coy Q j of Q; l D = e. (c) l forms an angle of at least α/2 with each of the tangents to D 1, D 2 at, and the edge e = a 1 a 2 D 1 D 2. Fix an arbitrary index 1 j h 2, so u j [] intersects e and forms an angle of at least α with each of w 1, w 2. Let Q j := Q (u j ) be the v j -lacement of Q at that touches. To see that such a lacement exists, we note that, by the receding remark, it suffices to show that the angle between and u j [] is at most π/2 α/2; that is, to rule out the case where lies in one of the shaded wedges in Figure 6 (c). This case is indeed imossible, because then one of u j 1 [], u j+1 [] would form an angle greater than π/2 with, contradicting the assumtion that both of these rays intersect the (Euclidean) b. The claim now follows from the next lemma, which shows that Q j D 1 D 2, which imlies that int Q j P = and thus Nj [] =, as claimed. Lemma 3.2. In the notation in the roof of Lemma 3.1, Q j D 1 D 2, for 1 j h 2. Proof. Fix a value of 1 j h 2. Let e be the edge of Q j assing through ; see Figure 7 (b). Let D be the disk whose center lies on u j [] and which asses through and, and let D + be the circumscribing disk of Q j. Since D D +, D int D +, and D and D + are centered on the ray u j [] emanating from, it follows that D D +. The line l containing e crosses D in a chord that is fully contained in e, as = D l D + l = e. The angle between the tangent to D at, denoted by τ, and the chord is eual to the angle at which sees. This angle is smaller than the angle at which sees e, which in turn is eual to 9
α/2. Recall that u j [] makes an angle of at least α with each of w 1 and w 2, therefore τ forms an angle of at least α with each of the tangents to D 1, D 2 at. Combining this with the fact that the angle between τ and e is at most α/2, we conclude that e forms an angle of at least α/2 with each of these tangents; see Figure 7 (c). The line l marks two chords 1, 2 within the resective disks D 1, D 2. We claim that e is fully contained in their union 1 2. Indeed, the angle 1 is eual to the angle between l and the tangent to D 1 at, so 1 α/2. On the other hand, the angle at which sees e is α/2, which is no larger. This, and the symmetric argument involving D 2, are easily seen to imly the claim. Now consider the circumscribing disk D + of Q j. Denote the endoints of e as a 1 and a 2, where a 1 lies in 1 and a 2 lies in 2. Since the ray a 1 hits D + before hitting D 1, and the ray hits these circles in the reverse order, it follows that the second intersection of D 1 and D + (other than ) must lie on a ray from which lies between the rays a 1, and thus crosses e. See Figure 7 (c). Symmetrically, the second intersection oint of D 2 and D + also lies on a ray which crosses e. It follows that the arc of D + delimited by these intersections and containing is fully contained in D 1 D 2. Hence all the vertices of Q j (which lie on this arc) lie in D 1 D 2. This, combined with the fact, established in the receding aragrah, that e 1 2 imlies that Q j D 1 D 2. Next, we use Lemma 3.1 to rove its converse. Secifically, we rove the following lemma. Lemma 3.3. Let, P be a air of oints such that Nj [] = for at least five consecutive indices j {0,..., k 1}. Then for each of these indices, excet ossibly for the first two and the last two indices, we have N j [] =. Proof. Again, assume with no loss of generality that Nj [] = for 0 j h 1, with h 5. Suose to the contrary that, for some 2 j h 3, we have N j [] =. By assumtion, Ni [] =, for each 0 i h 1, and therefore we have ϕ i [, ] ϕi [, ] <, for each of these indices. In articular, we have ϕ j [, ] ϕ j [, ] <, so there exists r P for which ϕ j[, r] < ϕ j [, ]. Assume with no loss of generality that r lies to the left of the line from to. In this case we claim that ϕ j 1 [, r] < ϕ j 1 [, ] < and ϕ j 2 [, r] < ϕ j 2 [, ] <. Indeed, the boundedness of ϕ j 1 [, ] and ϕ j 2 [, ] has already been noted. Moreover, because r lies to the left of the line from to, the orientation of b r lies counterclockwise to that of b. This, and our assumtion that u j [] hits b r before hitting b, imlies that the oint b r b lies to the right of the (oriented) line through u j []; see Figure 8. Hence, any ray ρ emanating from counterclockwise to u j [] that intersects b must also hit b r before hitting b, so we have ϕ j 1 [, r] < ϕ j 1 [, ] and ϕ j 2 [, r] < ϕ j 2 [, ] (since j 2, both u j 1 [] and u j 2 [] intersect b ), as claimed. Now alying Lemma 3.1 to the oint set {,, r} and to the index set {j 2, j 1, j}, we get that ϕ j 1 [, r] < ϕ j 1 [, ]. But this contradicts the fact that N j 1 [] =. The case where r lies to the right of is handled in a fully symmetric manner, using the indices {j, j + 1, j + 2}. Combining Lemmas 3.1 and 3.3, we obtain the following stronger version of Theorem 1.1. Theorem 3.4. Let P be a set of n oints in R 2, α > 0 a arameter, and Q a regular k-gon with k 2π/α. Then the following roerties hold. (i) Every 4α-stable edge in DT(P) is an α-stable edge in DT (P). (ii) Every 6α-stable edge in DT (P) is also an α-stable edge in DT(P). 10
r u j [] u j 2 [] u j 1[] b r b Figure 8. Proof of Lemma 3.3. Proof. Let be a 4α-stable edge of DT(P). Then the corresonding edge e in VD(P) stabs at least four rays u j [] emanating from, and, by Lemma 3.1, Nj [] = for at least two of these values of j. Therefore, sees the edge e in VD (P) at an angle at least α. Similarly sees the edge e at an angle at least α. Conversely, if is 6α-stable in DT (P) then e meets at least six rays u j [], and then Lemma 3.3 is easily seen to imly that (and, symmetrically, too) sees e at an angle at least α. The next lemma gives a slightly different characterization of stable edges, which is more algorithmic and which will be useful in maintaining a SDG under a constant-degree algebraic motion of the oints of P. Lemma 3.5. Let G be the subgrah of DT (P) comosed of the edges whose dual Q-Voronoi edges contain at least eleven breakoints. Then G is an (8α, α)-sdg of P (in the Euclidean norm). Proof. Let, P be two oints. If (, ) is an 8α-stable edge in DT(P) then the dual Voronoi edge e stabs at least eight rays u j [] emanating from, and at least eight rays u j [] emanating from. Lemma 3.1 imlies that VD (P) contains the edge e with at least six breakoints corresonding to corner lacements of Q at that touch, and at least six breakoints corresonding to corner lacements of Q at that touch. Therefore, e contains at least twelve breakoints, so (, ) G. Conversely, suose, P define an edge e in VD (P) with at least eleven breakoints. By the interleaving roerty of breakoints, stated in Lemma 2.1, we may assume, without loss of generality, that at least six of these breakoints corresond to emty corner lacements of Q at that touch. Lemma 3.3 imlies that VD(P) contains the edge e, and that this edge is hit by at least two consecutive rays u j []. But then the (, ) is α-stable in DT(P). In a comanion aer [3], we describe a kinetic data structure (KDS) for maintaining DT (P). As shown in that aer, it can also kee track of the number of breakoints for each edge of DT (P). If each oint of P moves along an algebraic trajectory of bounded degree, then the KDS rocesses O(k 4 nλ r (n)) events, where r is a constant deending on the comlexity of motion of P. A change in the number of breakoints in a Voronoi edge is an event that the KDS can detect and rocess. As discussed in detail in [3], many events, so-called singular events, that occur when an edge of DT (P) becomes arallel to an edge of Q, can occur simultaneously. Nevertheless, each of the events can be rocessed in O(log n) time, and their overall number is within the bound cited above. We maintain the subgrah G of DT (P), consisting of the edges of DT (P) that have at least eleven breakoints, which, by Lemma 3.5, is an (8α, α)-euclidean SDG. Putting everything together, we 11
obtain a KDS that maintains an (8α, α)-sdg of P. It uses linear storage, it rocesses O ( 1 α 4 nλ r (n) ) events, where r is a constant that deends on the degree of the motion of the oints of P, and it udates the SDG at each event in O(log n) time. This roves Theorem 1.2. Remarks. (i) We remark that DT (P) can undergo Ω(n) changes at a time instance when Ω(n) singular events occur simultaneously, say, when becomes arallel to an edge of Q, but all these changes occur at the edges incident to or in DT(P). However, only O(1/α) edges among them can have at least eleven breakoints, before or after the event. Hence, O(1/α) edges can simultaneously enter or leave the (8α, α)-sdg G of Theorem 1.2. (ii) Note that there is a slight discreancy between the value of k that we use in this section (k 2π/α), and the value needed to ensure that the regular k-gon Q k is α-close to the Euclidean disk, which is k π/α. This is made for the convenience of resentation. (iii) An interesting oen roblem is whether the deendence on α can be imroved in the above KDS. We have develoed an alternative scheme for maintaining stable (Euclidean) Delaunay grahs, which is reminiscent of the kinetic schemes used by Agarwal et al. [4] for maintaining closest airs and nearest neighbors. It extracts a nearly linear number of airs of oints of P that are candidates for being stable Delaunay edges and then sifts the stable edges from these candidate airs using the so-called kinetic tournaments [10]. Although the overall structure is not comlicated, the analysis is rather technical and lengthy, so we omit this KDS from this version of the aer; it can be found in the arxiv version [2]. In summary, the resulting KDS is of size O((n/α 2 ) log n), it rocesses a total of O((n/α) 2 β r (n/α) log 2 n log(log(n)/α)) events, and it takes a total of O((n/α) 2 log 2 n(log 2 n + β r (n/α) log 2 n log 2 (log(n)/α))) time to rocess them; here β r (n) = λ r (n)/n is an extremely slowly growing function for any fixed r. The worst-case time of rocessing an event is O(log 4 (n)/α)). Another advantage of this data structure is that, unlike the above KDS, it is local in the terminology of [10]. Secifically, each oint of P is stored, at any given time, at only O(log 2 (n)/α 2 ) laces in the KDS. Therefore the KDS can efficiently accommodate an udate in the trajectory of a oint. 4 Stability under Nearly Euclidean Distance Functions In this section we rove Theorem 1.1 for an arbitrary convex distance function that is α-close to the Euclidean norm (see the Introduction for the definition). Let Q be a comact, convex set that contains the origin in its interior, and let d Q denote the distance function induced by Q. Assume that Q is α-close to the Euclidean norm. For sake of brevity, we carry out the roof assuming that Q is strictly convex, i.e., the relative interior of the chord connecting two oint x and y on Q is strictly contained in the interior of Q (there are no straight segments on the boundary of Q). The roof also holds verbatim when Q is not strictly convex, rovided that no air of oints, P is such that is arallel to a straight ortion of Q. We assume that P is in general osition with resect to Q, in the sense that no three oints of P lie on a line, and no four oints of P are Q-cocircular, i.e., no four oints of P lie on the boundary of a common homothetic coy of Q. Recall that for a direction u and for a oint P, Q (u) denotes a generic homothetic coy of Q that touches and is centered at some oint on u[]. See Figure 4 (a). As mentioned in Section 2, all homothetic coies of Q (u) touch at (oints corresonding to) the same oint ζ Q and 12
therefore share the same tangent l (u) at. If Q is not smooth at ζ, there is a range of ossible orientations of such tangents. In this situation we let l (u) denote an arbitrary tangent of this kind. In addition, if Q is smooth at ζ then Q (u) exists for any oint that lies in the same side of l (u) as Q (u). Otherwise, this has to hold for every ossible tangent l (u) at ζ, which is euivalent to reuiring that lies in the wedge formed by the intersection of the two halflanes bounded by the two extreme tangents at ζ and containing Q (u). Remarks. (1) An imortant observation is that, when satisfies these conditions, Q (u) is uniue, unless all the three following conditions hold: (i) Q is not strictly convex, (ii) is arallel to straight ortion e of Q, and (iii) u is a direction connecting some oint on e to the center c of Q. We leave the straightforward roof of this roerty to the reader. The roof of the theorem exloits the uniueness of Q (u), and breaks down when it is not uniue. In fact, this is the only way in which the assumtions concerning strict convexity are used in the roof. (2) With some care, our analysis alies also if (the directions of) some airs are arallel to straight ortions of Q, in which case Q (u) is not uniuely defined for certain directions u. This extension reuires the more elaborate notion of α-stability, which ignores the ossible twodimensional ortions of e Q ; see Section 2 for more details. Informally, this allows us to avoid the roblematic directions u in which Q (u) is not uniue. (The latter haens exactly when u[] hits b Q within one of its two-dimensional ortions.) We note, though, that the loss in the amount of stability caused by ignoring a two-dimensional ortion of e Q is at most 2α. which is an uer bound on the angular san of directions that connect a straight ortion of Q to its center. This latter roerty holds since Q is α-close to the Euclidean disk; see below for more details. The roof of Theorem 1.1 relies on the following three simle geometric roerties. Recall that the α-closeness of Q to the Euclidean norm means that D I = (cos α)d O Q D O. τ θ y x θ o γ l l x τ 1 τ 2 x y x y o l y D O Q D O Q (a) (b) Figure 9. Illustrations for: (a) Claim 4.1; (b) Claim 4.2. Claim 4.1. Let x be a oint on Q, and let l be a suorting line to Q at x. Let y be the oint on D O closest to x (x and y lie on the same radius from the center o), and let γ be the arc of D O that contains y and is bounded by the intersection oints of l with D O. Then the angle between l and the tangent, τ, to D O at any oint along γ, is at most α. Proof. Denote this angle by θ. Clearly θ is maximized when τ is tangent to D O at an intersection of 13
l and D O ; see Figure 9 (a). For this value of θ, it is easy to verify that the distance from o to l is cos θ. But this distance has to be at least cos α, because l fully contains D I = (cos α)d O on one side. Hence cos θ cos α, and thus θ α, as claimed. Remark. An easy conseuence of this claim is that the angle in which the center of Q sees any straight ortion of Q (when Q is not strictly convex) is at most 2α. Claim 4.2. Let x and y be two oints on Q, and let l x and l y be suorting lines of Q at x and y, resectively. Then the difference between the (acute) angles that l x and l y form with xy is at most 2α. Proof. Denote the two angles in the claim by θ x and θ y, resectively. Continue the segment xy beyond x and beyond y until it intersects D O at x and y, resectively. Let τ 1 and τ 2 denote the resective tangents to D O at x and at y. See Figure 9 (b). Clearly, the resective angles θ 1, θ 2 between the chord x y of D O and τ 1, τ 2 are eual. By Claim 4.1 (alied once to τ 1 and l x and once to τ 2 and l y ) we get that θ 1 θ x α and θ 2 θ y α, and the claim follows. D O o γ z Q l Figure 10. Proof of Claim 4.3. Claim 4.3. For a oint Q, any tangent l to Q at forms an angle at most α with any line orthogonal to o. Proof. See Figure 10. Consider the chord ξ = l D O, and let γ denote the arc of D O determined by ξ and containing the intersection oint z of D O and the ray o. By Claim 4.1, the angle between l and the tangent to D O at z is at most α. Since this tangent is orthogonal to o, the claim follows. Remark. Clearly, Claims 4.1 4.3 continue to hold for any homothetic coy of Q, with a corresonding translation and scaling of D O and D I. Let Q (u) (res., Q + (u)) denote the ortion of Q (u) that lies to the left (res., right) of the directed line from to. Let D (u) denote the disk that touches and, and whose center lies on u[]. We next establish the following lemma, whose setu is illustrated in Figure 11 (a). It rovides the main geometric ingredient for the roof of Theorem 1.1. Lemma 4.4. (i) Let u S 1 be a direction such that both Q (u) and D (u + 5α) are defined. Then the region Q + (u) is fully contained in the disk D (u + 5α). (ii) Let u S 1 be a direction such that both Q (u) and D (u 5α) are defined. Then the region Q (u) is fully contained in the disk D (u 5α). 14
Proof. It suffices to establish Part (i) of the lemma; the roof of the other art is fully symmetric. Refer to Figure 11 (b). Let l = l (u) be any suorting line of Q (u) at, as defined above, and let τ = τ (u) be the line through that is orthogonal to u (which is also the tangent to D (u)). By Claim 4.3, the angle between l and τ is at most α. We next consider the tangent τ + to D (u + 5α) at. Since the angle between τ and τ + is 5α, it follows that the angle between l and τ + is at least 4α. The receding arguments imly that, when oriented into the right side of, l lies between and τ +, and the angle between l and τ + is at least 4α. This imlies that, locally near, Q + (u) enetrates into D (u + 5α). This also holds at. To establish the claim for (which is not symmetric to the claim for, because the center c of Q (u) lies on the ray u[] emanating from, and there is no control over the orientation of the corresonding ray c emanating from ), we note that, by Claim 4.2, the angles between and any air of tangents l, l to Q (u) at,, resectively, differ by at most 2α, whereas the angles between and the two tangents τ +, τ + to D (u + 5α) at,, resectively, are eual. This, and the fact that the angle between τ + and l is at least 4α, imly that, when oriented into the right side of, l lies between and τ +, which thus imlies the latter claim. Note also that the argument just given ensures that the angle between τ + and l is at least 2α. τ + (u 5α)[] u[] (u + 5α)[] Q (u) u D (u + 5α) (u + 5α) l τ w + l w w Q (u) Q + (u) D (u 5α) D (u + 5α) τ τ + l (a) (b) Figure 11. (a) The setu of Lemma 4.4; (b) Proof of Lemma 4.4 (i): Q + (u) cannot cross D (u + 5α) at any third oint w. It therefore suffices to show that Q + (u) does not cross D (u + 5α) at any third oint (other than and ). Suose to the contrary that there exists such a third oint w, and consider the tangents l w to Q (u) at w, and τ w + to D (u + 5α) at w. Consider the two oints and w, and aly to them an argument similar to the one used above for and. Secifically, we use the facts that (i) the angles between w and τ +, τ w + are eual, (ii) the angles between w and l, l w, for any tangent l w to Q (u) at w, differ by at most 2α, and (iii) the angle between l and τ + is at least 4α, to conclude that, when oriented into the left side of w, l w lies strictly between w and τ w +. See Figure 11. Similarly, alying the receding argument to and w, we now use the facts that (i) the angles between w and τ +, τ w + are eual, (ii) the angles between w and l, l w differ by at most 2α, and (iii) the angle between τ + and l is at least 2α, to conclude that, when oriented into the right side of w, l w lies between w and τ w + or coincides with τ w +. This imossible configuration shows that w cannot exist, and conseuently that Q + (u) D (u + 5α). 15
Proof of Theorem 1.1 Part (i). Let be an 11α-stable edge in the Euclidean Delaunay triangulation DT(P). That is, the Euclidean Voronoi edge e is hit by two rays u [], u + [] which form an angle of at least 11α between them (where u + is assumed to lie counterclockwise to u ). Clearly, e is also hit by any ray u[] whose direction u belongs to the interval (u, u + ) S 1. Let u[] be such a ray whose direction u belongs to the interval (u + 5α, u + 5α) (of san at least α). That is, u[] hits e somewhere in the middle, so all the three disks D (u 5α), D (u) and D (u + 5α) are defined and contain no oints of P in their resective interiors. (Actually, D (u) is contained in D (u 5α) D (u + 5α), as is easily checked.) We next consider the Q-Voronoi diagram VD Q (P). We claim that the corresonding edge e Q exists and is also hit by u[]. Since this holds for every u (u + 5α, u + 5α), it follows that (, ) is α-stable in VD Q (P). To establish this claim, we rove the following two roerties. (i) the homothetic coy Q (u) exists, and (ii) it contains no oints of P in its interior. Proof of (i): Assume to the contrary that the coy Q (u) is undefined. Consider the resective tangents l (u) and τ (u) to Q (u) and D (u) at, where l (u) is any tangent to Q (u) at that searates from Q (u); such a tangent exists if and only if Q (u) is undefined. (As noted before, l (u) does not deend on the location of the center c of Q on u[].) By Claim 4.3, the angle between l (u) and τ (u) is at most α. Since Q (u) is undefined, the choice of l (u) guarantees that lies inside the oen halflane h (u) bounded by l (u) and disjoint from u[]. Let µ (u + 5α) (res., µ (u 5α)) denote the oen halflane bounded by τ (u + 5α) (res., τ (u 5α)) and disjoint from the disk D (u + 5α) (res., D (u 5α)). Since each of the lines τ (u + 5α), τ (u 5α) makes an angle of at least 5α with τ (u), the halflane h (u) suorted by l (u) is contained in the union µ (u + 5α) µ (u 5α). Since is contained in h (u), at least one of these latter halflanes, say µ (u + 5α), must contain. However, if µ (u + 5α), the corresonding coy D (u + 5α) is undefined, a contradiction that establishes (i). Proof of (ii): Since both Q (u) and D (u + 5α) are defined, Lemma 4.4(i) imlies that Q + (u) D (u + 5α). Moreover the interior of D (u + 5α) is P-emty, so the interior of Q + (u) is also P-emty. A symmetric argument (using Lemma 4.4(ii)) imlies that the interior of Q (u) is also P-emty. This comletes the roof of art (i) of Theorem 1.1. Proof of Theorem 1.1 Part (ii). We fix a direction u S 1 for which all the three coies Q (u), Q (u 5α), and Q (u + 5α) are defined and have P-emty interiors. Again, Q (u) Q (u 5α) Q (u + 5α). Since is 11α-stable under d Q, there is an arc on S 1 of length at least α, so that every u in this arc has this roerty. We need to show that, for every such u, (i) the coy D (u) is defined, and (ii) its interior is P-emty. Similar to the roof of Part (i), this would imly that the ray u[] hits the edge e of VD(P) for every u in an arc of length α, so is α-stable in DT(P), as claimed. 16
Proof of (i): Assume to the contrary that D (u) is undefined, so the angle between the vectors and u is at least π/2. Let l (u 5α), l (u), and l (u + 5α) be any trile of resective tangents to Q (u 5α), Q (u), and Q (u + 5α) at. Let h (u 5α) (res., h (u + 5α)) be the oen halflane suorted by l (u 5α) (res., l (u + 5α)) and disjoint from Q (u 5α) (res., Q (u + 5α)). Claim 4.3 imlies that each of the lines l (u 5α), l (u + 5α) makes with τ (u), the line orthogonal to u[] at, an angle of at least 4α (and at most 6α). Indeed, the claim imlies that the angle between l (u 5α) and the line τ (u 5α), which is orthogonal to (u 5α)[] at, is at most α. Since the angle between τ (u 5α) and τ (u) is 5α, the claim for l (u 5α) follows. A symmetric argument establishes the claim for l (u + 5α). Therefore, the halflane µ (u), suorted by τ (u) and containing, is covered by the union of h (u 5α) and h (u + 5α). We conclude that at least one of these latter halflanes must contain. However, this contradicts the assumtion that both coies Q (u 5α), Q (u + 5α) are defined, and (i) follows. Proof of (ii): Assume to the contrary that D (u), whose existence has just been established, contains some oint r of P in its interior. That is, the ray u[] hits b r before b. In this case D r (u) also exists. With no loss of generality, we assume that r lies to the left of the oriented line from to. We claim that the homothetic coy Q r (u 5α) exists and contains. Indeed, since Q (u 5α) exists and is P-emty, it follows that (u 5α)[] either hits b Q r after b Q (in which case the claim obviously holds) or misses b Q r altogether. Suose that (u 5α)[] misses b Q r. As argued earlier, this means that there exists a tangent l (u 5α) to Q (u 5α) at, such that r lies in the oen halflane h (u 5α) suorted by l (u 5α) and disjoint from Q (u 5α). By alying Claim 4.3 as before we get that the tangent τ (u) to D (u) (at ) to the left of is between and l (u 5α) and makes with l (u 5α) an angle of at least 4α. It follows that the wedge formed by the intersection of l (u 5α) and the halflane to the left of is fully contained in the halflane µ (u) that is suorted by τ (u) and disjoint from D (u); see Figure 12 (a). But then D r (u) is undefined, a contradiction that imlies the existence of Q r (u 5α). (u 5α)[] u[] D (u) (u 5α)[] r u[] l (u 5α) Q (u 5α) τ (u) D r (u) µ (u) r h (u 5α) Q r (u 5α) (a) (b) Figure 12. Proof of Theorem 1.1(ii): (a) Arguing (by contradiction) that Q r (u 5α) exists; (b) the coy Q r (u 5α) is defined and contains. Hence, the disk D r (u), which covers Q + r(u 5α), must contain as well. We can now assume that Q r (u 5α) is defined and contains. More recisely, lies in the ortion Q + r(u 5α) of Q r (u 5α), since lies to the right of the oriented line from to r. However, 17