Kinetic Stable Delaunay Graphs

Transcription

1 Kinetic Stable Delaunay Grahs Pankaj K. Agarwal Duke University Haim Kalan Tel Aviv University Jie Gao SUNY Stony Brook Vladlen Koltun Stanford University Micha Sharir Tel Aviv University and New York University Leonidas J. Guibas Stanford University Natan Rubin Tel Aviv University ABSTRACT The best known uer bound on the number of toological changes in the Delaunay triangulation of a set of moving oints in R 2 is (nearly) cubic, even if each oint is moving with a fixed velocity. We introduce the notion of a stable Delaunay grah (SDG in short), a dynamic subgrah of the Delaunay triangulation, that is less volatile in the sense that it undergoes fewer toological changes and yet retains many useful roerties of the full Delaunay triangulation. SDG is defined in terms of a arameter α > 0, and consists of Delaunay edges for which the (eual) angles at which and see the corresonding Voronoi edge e are at least α. We rove several interesting roerties of SDG and describe two kinetic data structures for maintaining it. Both structures use O (n) storage. They rocess O (n 2 ) events during the motion, each in O (1) time, rovided that the oints of P move along algebraic trajectories of bounded degree; the O ( ) notation hides multilicative factors that are olynomial in 1/α and olylogarithmic in n. The first structure is simler but the deendency on 1/α in its erformance is higher. Categories and Subject Descritors F.2.2 [Analysis of algorithms and roblem comlexity]: Nonnumerical algorithms and roblems Geometrical roblems and comutations; G.2.1 [Discrete mathematics]: Combinatorics Combinatorial algorithms General Terms Algorithms, Theory Keywords Delaunay triangulation, Voronoi diagram, kinetic data structures 1. INTRODUCTION Delaunay triangulation and Voronoi diagram. Let P be a (finite) set of oints in R 2. Let VD(P ) and DT(P ) denote the Voronoi di- Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, reuires rior secific ermission and/or a fee. SCG 10, June 13 16, 2010, Snowbird, Utah, USA. Coyright 2010 ACM /10/06...$ agram and Delaunay triangulation of P, resectively. For a oint P, let Vor() denote the Voronoi cell of. The Delaunay triangulation DT = DT(P ) consists of all triangles whose circumcircles do not contain oints of P in their interior. Its edges form the Delaunay grah, which is the straight-edge dual grah of the Voronoi diagram of P. That is, is an edge of the Delaunay grah if and only if Vor() and Vor() share an edge, which we denote by e. This is euivalent to the existence of a circle assing through and that does not contain any oint of P in its interior any circle centered at a oint on e and assing through and is such a circle. Delaunay triangulations and Voronoi diagrams are fundamental to much of comutational geometry and its alications. See [6, 11] for a survey and a textbook on these structures. In many alications of Delaunay/Voronoi methods (e.g., mesh generation and kinetic collision detection) the oints are moving continuously, so these diagrams need to be efficiently udated as motion occurs. Even though the motion of the nodes is continuous, the combinatorial and toological structure of the Voronoi and Delaunay diagrams change only at discrete times when certain critical events occur. Their evolution under motion can be studied within the framework of kinetic data structures (KDS in short) of Basch et al. [7, 12, 13], a general methodology for designing efficient algorithms for maintaining such combinatorial attributes of mobile data. For the urose of kinetic maintenance, Delaunay triangulations are nice structures, because, as mentioned above, they admit local certifications associated with individual triangles. This makes it simle to maintain DT under oint motion: an udate is necessary only when one of these emty circumcircle conditions fails this corresonds to co-circularities of certain subsets of four oints. Whenever such an event haens, a single edge fli easily restores Delaunayhood. Estimating the number of such events, however, has been elusive the roblem of bounding the number of combinatorial changes in DT for oints moving with constant velocities has been in the comutational geometry lore for many years. Let n be the number of moving oints in P. We assume that each oint moves along an algebraic trajectory of fixed degree (see Section 2 for a more formal definition). Guibas et al. [14] showed a roughly cubic uer bound of O(n 2 λ s (n)) on the number of discrete (also known as toological) changes in DT, where λ s(n) is the maximum length of an (n, s)-davenort-schinzel seuence [21], and s is a constant deending on the motions of the oints. A substantial ga exists between this uer bound and a uadratic lower bound [21]. It is thus desirable to find aroaches for maintaining a substantial ortion of DT that rovably exeriences only a nearly uadratic

2 number of discrete changes, that is reasonably easy to define and maintain, and that retains useful roerties for further alications. Polygonal distance functions. If the unit ball" of our underlying norm is olygonal then things imrove considerably. In more detail, let Q be a convex olygon with a constant number, k, of edges. It induces a convex distance function d Q(x, y) = min{λ y x + λq}; d Q is a metric if Q is centrally symmetric with resect to the origin. We can define the Q-Voronoi diagram of a set P of oints in the lane in the usual way, as the artitioning of the lane into Voronoi cells, so that the cell Vor () of a oint is {x R 2 d Q (x, ) = min P d Q (x, )}. Assuming that the oints of P are in general osition with resect to Q, these cells are nonemty, have airwise disjoint interiors, and cover the lane. As in the Euclidean case, the Q-Voronoi diagram of P has its dual reresentation, which we refer to as the Q-Delaunay triangulation DT (P ) = DT. A trile of oints in P define a triangle in DT if and only if they lie on the boundary of some homothetic coy of Q that does not contain any oint of P in its interior. Assuming that P is in general osition, these Q-Delaunay triangles form a triangulation of a certain simly-connected olygonal region that is contained in the convex hull of P. Unlike the Euclidean case, it does not always coincide with the convex hull (see Figures 3 and 11 for examles). See Chew and Drysdale [9] and Leven and Sharir [18] for analysis of Voronoi and Delaunay diagrams of this kind. For kinetic maintenance, olygonal Delaunay triangulations are better than Euclidean Delaunay triangulations because, as shown by Chew [8], when the oints of P move (in the algebraic sense assumed above), the number of toological changes in the Q-Delaunay triangulation is only nearly uadratic in n. One of the major observations in this aer is that the stable ortions of the Euclidean Delaunay triangulation and the Q-Delaunay triangulation are closely related. 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. (The case where lies on the convex hull of P is treated as if one of r +, r lies at infinity, so that the corresonding angle r + or 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 eual angles at which and see their common Voronoi edge e are at least α. See Figure 1 (left). A justification for calling such edges stable lies in the following observation: If a Delaunay edge is α-stable then it remains in DT during any continuous motion of P for which every angle r, for r P \{, }, changes by at most α/2. This is clear because at the time is α-stable we have r + + r < π α for any air of oints r +, r lying on oosite sides of the line l suorting, so, if each of these angles change by at most α/2 we still have r + + r π, which is easily seen to imly that remains an edge of DT. (This argument also covers the cases when a oint crosses l from side to side.) 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 are those for and are almost co-circular with their two common Delaunay neighbors r +, r, and hence more likely to get flied soon". Overview of our results. Let α > 0 be a fixed arameter. In this aer we show how to maintain a subgrah of the full Delaunay triangulation DT, which we call a (cα, α)-stable Delaunay grah (SDG in short), so that (i) every edge of SDG is α-stable, and (ii) every cα-stable edge of DT belongs to SDG, where c > 1 is some (small) absolute constant. Note that SDG is not uniuely defined, even when c is fixed. In Section 2, we introduce several useful definitions and show that the number of discrete changes in the SDGs that we consider is nearly uadratic. What this analysis also imlies is that if the true bound for kinetic changes in a Delaunay triangulation is close to cubic, then the overhelming majority of these changes involve edges which never become stable and just flicker in and out of the diagram by co-circularity with their two Delaunay neighbors. In Sections 3 and 4 we show that SDG can be maintained by a kinetic data structure that uses only near-linear storage (in the terminology of [7], it is comact), encounters only a nearly uadratic number of critical events (it is efficient), and rocesses each event in olylogarithmic time (it is resonsive). For the second data structure described in Section 4, each oint aears at any time in only olylogarithmically many laces in the structure (it is local). The scheme described in Section 3 is based on a useful and interesting euivalence" connection between the (Euclidean) SDG and a suitably defined stable" version of the Delaunay triangulation of P under the olygonal" norm whose unit ball Q is a regular k-gon, for k = Θ(1/α). As noted above, Voronoi and Delaunay structures under olygonal norms are articularly favorable for kinetic maintenance because of Chew s result [8], showing that the number of toological changes in DT (P ) is O (n 2 k 4 ); the O ( ) notation hides a factor that deends sub-olynomially on both n and k. In other words, the scheme simly maintains the olygonal" diagram DT (P ) in its entirety, and selects from it those edges that are also stable edges if the Euclidean diagram DT. The major disadvantage of the solution in Section 3 is the rather high (roortional to Θ(1/α 4 )) deendence on 1/α( k) of the bound on the number of toological changes. We do not know whether the uer bound O (n 2 k 4 ) on the number of toological changes in DT (P ) is nearly tight (in its deendence on k). To remedy this, we resent in Section 4 an alternative scheme for maintaining stable (Euclidean) Delaunay edges. The scheme is reminiscent of the kinetic schemes used in [2] for maintaining closest airs and nearest neighbors. It extracts O (n) airs of oints of P that are candidates for being stable Delaunay edges. Each oint P then runs O(1/α) kinetic and dynamic tournaments involving the other oints in its candidate airs. Roughly, these tournaments corresond to shooting O(1/α) rays from P in fixed directions and finding along each ray the nearest oint eually distant from and from some other candidate oint. We show that is a stable Delaunay edge if and only if wins many consecutive tournaments of (or wins many consecutive tournaments of ). A careful analysis shows that the number of events that this scheme rocesses (and the overall rocessing time) is only O (n 2 /α 2 ). Section 5 reviews several useful roerties of stable Delaunay grahs. In articular, we show that at any given time the stable subgrah contains at least [ 1 3 2(π/α 2) ] n Delaunay edges, i.e., at least about one third of the maximum ossible number of edges. In addition, we show that at any moment the SDG contains the closest air, the so-called -skeleton of P, for = 1 + Ω(α 2 ) (see [5, 17]), and the crust of a sufficiently densely samled oint set along a smooth curve (see [4, 5]). We also extend the connection in Section 3 to arbitrary distance functions d Q whose unit ball Q is sufficiently close (in the Hausdorff sense) to the Euclidean one (i.e.,

3 the unit disk). Comleting DT (P ) into a triangulation of the entire hull is an interesting challenge that we are currently exloring in a comanion aer [16]. 2. PRELIMINARIES Stable edges in Voronoi diagrams. Let {u 0,..., u k 1 } S 1 be a set of k = Θ(1/α) eually saced directions in R 2. For concreteness take u i = (cos(2πi/k), sin(2πi/k)), 0 i < k (so our directions u i go clockwise as i increases). 1 For a oint P and a (unit) vector u let u[] denote the ray {+λu λ 0} that emanates from in direction u. For a air of oints, P let b denote the erendicular bisector of and. If b intersects u i[], then the exression ϕ i[, ] = 2 2, u i is the distance between and the intersection oint of b with u i[]. If b does not intersect u i[] we define ϕ i[, ] =. The oint minimizes ϕ i [, ], among all oints for which b intersects u i [], 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(); see Figure 1 (right). The (angular) extent of a Voronoi edge e of two oints, P is the angle at which it is seen from either or (these two angles are eual). For a given angle α π, e is called α-long (res., α-short) if the extent of e is at least (res., smaller than) α. We also say that DT(P ) is α-long (res., α-short) if e is α-long (res., α-short). As noted in the introduction, these notions can also be defined (euivalently) in terms of the angles in the Delaunay triangulation: A Delaunay edge is α-long if and only if r + + r π α, where r + and r are the two Delaunay triangles incident to. See Figure 1(left). Given arameters α > α > 0, we seek to construct (and maintain under motion) an (α, α)-stable Delaunay grah (or stable Delaunay grah, for brevity, which we abbreviate as SDG) of P, which is any subgrah G of DT(P ) with the following roerties: (S1) Every α -long edge of DT(P ) is an edge of G. (S2) Every edge of G is an α-long edge of DT(P ). An (α, α)-stable Delaunay grah need not be uniue. In what follows, α will always be some fixed (and reasonably small) multile of α. Kinetic tournaments. Kinetic tournaments were first studied by Basch et al. [7], for kinetically maintaining the lowest oint in a set P of n oints moving on some vertical line, say the y-axis, so that their trajectories are algebraic of bounded degree, as above. We also want the tournament to be dynamic, so that we can insert and delete oints into/from it. This can be achieved by maintaining the tournament as a hea, stored as a a weight-balanced (BB(α)) tree [20] (and [19]). Omitting all further details, which can be found in [2] (and in the full version [1]), we state: THEOREM 2.1 (AGARWAL et al. [2]). A seuence of m insertions and deletions into a kinetic tournament, whose maximum size at any time is n (assuming m n), when imlemented as a weight-balanced tree, generates at most O(m r+2(n) log n) events, with a total rocessing cost of O(m r+2(n) log 2 n). Here r is the maximum number of times a air of oints intersect, and r+2 (n) = λ r+2 (n)/n. Processing an udate or a tournament event takes O(log 2 n) worst-case time. A dynamic kinetic tournament on n elements can be constructed in O(n) time. 1 The index arithmetic is modulo k, i.e., u i = u i+k. (1) Maintenance of an SDG. Let P = { 1,..., n } be a set of oints moving in R 2. Let i (t) = (x i (t), y i (t)) denote the osition of i at time t. We call the motion of P algebraic if each x i( ), y i( ) is a olynomial, and the degree of motion of P is the maximum degree of these olynomials. Throughout this aer we assume that the motion of P is algebraic and that its degree is bounded by a constant. In this subsection we resent a simle techniue for maintaining a (2α, α)-stable Delaunay grah. Unfortunately this algorithm reuires uadratic sace. It is based on the following easy observation (see Figure 1 (right)), where k is an integer, and the unit vectors (directions) u 0,..., u k 1 are as defined earlier. LEMMA 2.2. Let α = 2π/k. (i) If the extent of e is larger than 2α then there are two consecutive directions u i, u i+1, such that is the neighbor of in directions u i and u i+1. (ii) If there are two consecutive directions u i, u i+1, such that is the neighbor of in both directions u i and u i+1, then the extent of e is at least α. r y r + a 2y x 2x b b Vor() α u i ui+1 Figure 1. Left: The oints and see their common Voronoi edge ab at (eual) angles. This is euivalent to the angle condition x + y = π for the two adjacent Delaunay triangles. Right: is the neighbor of in the directions u i and u i+1, so the Voronoi edge e is α-long. The algorithm maintains Delaunay edges such that there are two consecutive directions u i and u i+1 along which is the neighbor of. For each oint and direction u i we get a set of at most n 1 iecewise continuous functions of time, ϕ i[, ], one for each oint, as defined in (1). (Recall that ϕ i[, ] = when u i [] does not intersect b.) By assumtion on the motion of P, for each and, the domain in which ϕ i [, ](t) is defined consists of a constant number of intervals. For each oint, and ray u i[], consider each function ϕ i[, ] as the trajectory of a oint moving along the ray and corresonding to. The algorithm maintains these oints in a dynamic and kinetic tournament K i () (see Theorem 2.1) that kees track of the minimum of {ϕ i [, ](t)} over time. For each air of oints and such that wins in two consecutive tournaments, K i() and K i+1 (), of, it kees the edge in the stable Delaunay grah. It is trivial to udate this grah as a by-roduct of the udates of the various tournaments. The analysis of this data structure is straightforward using Theorem 2.1, and yields the following result. THEOREM 2.3. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, let k be an integer, and let α = 2π/k. A (2α, α)-stable Delaunay grah of P can be maintained using O(kn 2 ) storage and rocessing O(kn 2 r+2 (n) log n) events, for a total cost of O(kn 2 r+2(n) log 2 n) time. The rocessing of each event takes O(log 2 n) worst-case time. Here r is a constant that deends on the degree of motion of P. 3. AN SDG BASED ON POLYGONAL VD Let Q = Q k be a regular k-gon for some even k = 2s, circumscribed by the unit disk, and let α = π/s (this is the angle at which

4 the center of Q sees an edge). Let VD (P ) and DT (P ) denote the Q-Voronoi diagram and the dual Q-Delaunay triangulation of P, resectively. In this section we show that the set of edges of VD (P ) with sufficiently many breakoints form a (, )-stable (Euclidean) Delaunay grah for aroriate multiles, of α. Thus, by kinetically maintaining VD (P ) (in its entirety), we shall get for free a KDS for keeing track of a stable ortion of the Euclidean DT. 3.1 Proerties of VD (P) We first review the roerties of the (stationary) VD (P ) and DT (P ). Then we consider the kinetic version of these diagrams, as the oints of P move, and review Chew s roof [8] that the number of toological changes in these diagrams, over time, is only nearly uadratic in n. Finally, we resent a straightforward kinetic data structure for maintaining DT (P ) under motion that uses linear storage, and that rocesses a nearly uadratic number of events, each in O(log n) time. Although later on we will take Q to be a regular k-gon, the analysis in this subsection is more general, and we only assume here that Q is an arbitrary convex k-gon. Stationary Q-diagrams. The bisector b between two oints and, with resect to d Q(, ), is the locus of all lacements of the center of any homothetic coy Q of Q that touches and. Q can be classified according to the air of its edges, e 1 and e 2, that touch and, resectively. If we slide Q so that its center moves along b (and its size exands or shrinks to kee it touching and ), and the contact edges, e 1 and e 2, remain fixed, the center traces a straight segment. The bisector is a concatenation of O(k) such segments. They meet at breakoints, which are lacements of the center of a coy Q that touches and and one of the contact oints is a vertex of Q; see Figure 2. We call such a lacement a corner contact at the aroriate oint. Note that a corner contact where some vertex w of (a coy Q of) Q touches has the roerty that the center of Q lies on the fixed ray emanating from and arallel to the directed segment from w to the center of Q. b Figure 2. Each breakoint on b corresonds to a corner contact of Q at one of the oints,, so that Q also touches the other oint. A well known roerty of Q-bisectors and Voronoi edges is that two bisectors b 1, b 2, can intersect at most once (again, assuming general osition) [15], so every Q-Voronoi edge e is connected. Euivalently, this asserts that there exists at most one homothetic lacement of Q at which it touches, 1, and 2. Another useful roerty of bisectors and Delaunay edges, in the secial case where Q is a regular k-gon, which will be used in the next subsection, is that the breakoints along a bisector b alternate between corner contacts at and corner contacts at. The roof is omitted here. Consider next an edge of DT (P ). Its dual Voronoi edge e is a ortion of the bisector b, and consists of those center lacements along b for which the corresonding coy Q has an emty interior (i.e., its interior is disjoint from P ). Following the notation of Chew [8], we call a corner edge if e contains a breakoint (i.e., a lacement with a corner contact); otherwise it is a non-corner edge, and is therefore a straight segment. Kinetic Q-diagrams. Consider next what haens to VD (P ) and DT (P ) as the oints of P move continuously with time. In this case VD (P ) changes continuously, but undergoes toological changes at certain critical times, called events. There are two kinds of events: (i) [FLIP EVENT.] A Voronoi edge e shrinks to a oint, disaears, and is flied into a newly emerging Voronoi edge e. (ii) [CORNER EVENT.] An endoint of some Voronoi edge e becomes a breakoint (a corner lacement). Immediately after this time e either gains a new straight segment, or loses such a segment. Some comments are in order: (a) A fli event occurs when the four oints,,, become cocircular : there is an emty homothetic coy Q of Q that touches all four oints. (b) Only non-corner edges can articiate in a fli event, as both the vanishing edge e and the newly emerging edge e do not have breakoints near the event. (c) A fli event entails a discrete change in the Delaunay triangulation, whereas a corner event does not. Still we will kee track of both kinds of events. We first bound the number of corner events. LEMMA 3.1. Let P be a set of n oints in R 2 under algebraic motion of bounded degree, and let Q be a convex k-gon. The number of corner events in DT (P ) is O(k 2 nλ r(n)), where r is a constant that deends on the degree of motion of P. PROOF. Fix a oint and a vertex w of Q, and consider all the corner events in which w touches. As noted above, at any such event the center c of Q lies on a ray γ emanating from at a fixed direction. For each other oint P \ {}, let ϕ γ[, ] denote the distance, at time t, from along γ to the center of a coy of Q that touches (at w) and. The value min ϕ γ[, ](t) reresents the intersection of Vor () with γ at time t, where Vor () is the Voronoi cell of in VD (P ). The oint that attains the minimum defines the Voronoi edge e (or vertex if the minimum is attained by more than one oint ) that intersects γ. In other words, we have a collection of n 1 artially defined functions ϕ γ[, ], and the breakoints of their lower enveloe reresent the corner events that involve the contact of w with. By our assumtion on the motion of P, each function ϕ γ[, ] is iecewise algebraic, with O(k) ieces. Each iece encodes a continuous contact of with a secific edge of Q, and has constant descrition comlexity. Hence (see, e.g., [21, Corollary 1.6]) the comlexity of the enveloe is at most O(kλ r (n)), for an aroriate constant r. Reeating the analysis for each oint and each vertex w of Q, the lemma follows. Consider next fli events. As noted, each fli event involves a lacement of an emty homothetic coy Q of Q that touches simultaneously four oints 1, 2, 3, 4 of P, in this counterclockwise order along Q, so that the Voronoi edge e 1 3, which is a non-corner edge before the event, shrinks to a oint and is relaced by the non-corner edge e 2 4. Let e i denote the edge of Q that touches i, for i = 1, 2, 3, 4. We fix the uadrule of edges e 1, e 2, e 3, e 4, bound the number of fli events involving a uadrule contact with these edges, and sum the bound over all O(k 4 ) choices of four edges of Q. For a fixed uadrule of edges e 1, e 2, e 3, e 4, we relace Q by the convex

5 hull Q 0 of these edges, and note that any fli event involving these edges is also a fli event for Q 0. We therefore restrict our attention to Q 0, which is a convex k 0-gon, for some k 0 8. The number of fli events for a fixed Q 0 is bounded by extending and adating the analysis of Chew [8]. Basically, we surround a non-corner edge by O(1) nearby corner edges, and argue that, as long as none of them undergoes a corner event, only O(1) fli events can occur in the middle" edge. This allows us to charge a fli event to a corner event so that each corner event is charged only O(1) times. See [8] and the full version [1] for details. Reeating this to each uadrule of edges of Q, we thus obtain: THEOREM 3.2. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, and let Q be a convex k-gon. The number of toological changes in VD (P ) with resect to Q is O(k 4 nλ r (n)), where r is a constant that deends on the degree of motion of P. Kinetic maintenance of VD (P) and DT (P). As already mentioned, it is a fairly trivial task to maintain DT (P ) and VD (P ) kinetically, as the oints of P move. All we need to do is to assert the correctness of the resent triangulation by a collection of local certificates, one for each edge of the diagram, where the certificate of an edge asserts that the two homothetic lacements Q, Q + of Q that circumscribe the two resective adjacent Q-Delaunay triangles r, r +, are such that Q does not contain r + and Q + does not contain r. The failure time of this certificate is the first time (if one exists) at which,, r, and r + become Q-cocircular they all lie on the boundary of a common homothetic coy of Q. If is an edge of the erihery of DT (P ), so that r + exists but r does not, then Q is a limiting wedge bounded by rays suorting two consecutive edges of (a coy of) Q, one assing through and one through (see Figure 3). The failure time of the corresonding certificate is the first time (if any) at which r + also lies on the boundary of that wedge. This corresonds to a fli event. We maintain the breakoints using sub-certificates", each of which asserts that Q, say, touches each of,, r at a resective secific edge (and similarly for Q + ). The failure time of this subcertificate is the first failure time when one of, or r touches Q at a vertex. In this case we have a corner event two of the adjacent Voronoi edges terminate at a corner lacement. Note that the failure time of each sub-certificate can be comuted in O(1) time. Moreover, for a fixed collection of valid sub-certificates, the failure time of an initial certificate (asserting non-co-circularity) can also be comuted in O(1) time (rovided that it fails before the failures of the corresonding sub-certificates), because we know the four edges of Q involved in the contacts. Q Figure 3. If r does not exist then Q is a limiting wedge bounded by rays suorting two consecutive edges of (a coy of) Q. We therefore maintain an event ueue that stores and udates all the active failure times (there are only O(n) of them the bound is indeendent of k). When a sub-certificate fails we do not change r + DT (P ), but only udate the corresonding Voronoi edge, by adding or removing a segment and a breakoint, and by relacing the subcertificate by a new one; we also udate the real certificate associated with the edge, because one of the contact edges has changed. When a real certificate fails we udate DT (P ) and construct O(1) new sub-certificates and certificates. Altogether, each udate of the diagram takes O(log n) time. We thus have THEOREM 3.3. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, and let Q be a convex k-gon. DT (P ) and VD (P ) can be maintained using O(n) storage and O(log n) udate time, so that O(k 4 nλ r(n)) events are rocessed, where r is a constant that deends on the degree of motion of P. 3.2 Stable Delaunay edges in DT (P) We now restrict Q to be a regular k-gon. Let v 0,..., v k 1 be the vertices of Q arranged in a clockwise direction, with v 0 the leftmost. We call a homothetic coy of Q whose vertex v j touches a oint, a v j-lacement of Q at. Let u j be the direction of the vector that connects v j with the center of Q, for each 0 j < k (as in Section 2). See Figure 4 (left). We follow the machinery in the roof of Lemma 3.1. That is, 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. If ϕ j [, ] < then the oint b u j [] is the center of the v j -lacement Q of Q at that also touches. The value ϕ j [, ] is eual to the circumradius of Q. See Figure 4 (middle). Q v j u j α [, ] ϕ j b u j[] u j[] Figure 4. Left: u j is the direction of the vector connecting vertex v j to the center of Q. Middle: The function ϕ j [, ] is eual to the radius of the circle that circumscribes the v j -lacement of Q at that also touches. Right: The case when ϕ j [, ] = while ϕ j[, ] <. In this case must lie in one of the shaded wedges. The neighbor N j [] of in direction u j is defined to be the oint P \ {} that minimizes ϕ j [, ]. Clearly, for any, P we have N j [] = if and only if there is an emty v j -lacement Q of Q at so that touches one of its edges. Remark: Note that, in the Euclidean case, we have ϕ j [, ] < if and only if the angle between and u j [] is at most π/2. In contrast, ϕ j [, ] < if and only if the angle between and u j[] is at most π/2 π/k = π/2 α/2. Moreover, we have ϕ j [, ] ϕ j [, ]. Therefore, ϕ j [, ] < always imlies ϕ j [, ] <, but not vice versa; see Figure 4 (right). LEMMA 3.4. Let, P be a air of oints such that N j () = for h 3 consecutive indices, say 0 j < h. Then for each of these indices, excet ossibly for the first and the last one, we also have N j [] =. 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

6 D 1 w1 w 2 D 2 l e u j[] Figure 5. Left: The angle between the tangents to D 1 and D 2 is eual to w 1 w 2 = = (h 1)α. Right: The line l crosses D in a chord which is fully contained in e (right). tangents to D 1 and D 2 at or at (these angles are eual) is = (h 1)α; see Figure 5 (left). 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 be the v j -lacement of Q at that touches. The existence of such a lacement follows from the receding remark. We claim that Q D 1 D 2. Establishing this roerty for every 1 j h 2 will comlete the roof of the lemma. Let e be the edge of Q assing through. 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. Since D and is interior to D +, and since D and D + are centered on the same ray u j [] and ass through, it follows that D D +. See Figure 5 (right). The line l containing e crosses D in a chord that is fully contained in e. The angle between the tangent to D at and the chord is eual to the angle at which sees. This is smaller than the angle at which sees e, which in turn is eual to α/2. Q D l 1 2 l a e 1 a 2 a e 1 a 2 D D+ D 1 D D 1 D 2 D 2 Figure 6. Left: The line l forms an angle of at least α/2 with each of the tangents to D 1, D 2 at. Right: The edge e = a 1 a 2 of Q is fully contained in D 1 D 2. Arguing as in the analysis of D 1 and D 2, the tangent to D at forms an angle of at least α with each of the tangents to D 1, D 2 at, and hence e forms an angle of at least α/2 with each of these tangents; see Figure 6 (left). 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 smaller. This, and the symmetic argument involving D 2, are easily seen to imly the claim. Now consider the circumscribing disk D + of Q. 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 6 (right). 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 D vertices of Q (which lie on this arc) lie in D 1 D 2. This, combined with the argument in the receding aragrahs, is easily seen to imly that Q D 1 D 2, so its interior does not contain oints of P, which in turn imlies that N j [] =. As noted, this comletes the roof of the lemma. Since Q-Voronoi edges are connected, Lemma 3.4 imlies that e is long", in the sense that it contains at least h 2 breakoints that reresent corner lacements at, interleaved (as romised in Section 3.1) with at least h 3 corner lacements at. This roerty is easily seen to hold also under the weaker assumtions that: (i) for the first and the last indices j = 0, h 1, the oint N j [] either is eual to or is undefined, and (ii) for the rest of the indices j we have N j [] = and ϕ j [, ] < (i.e., the v j -lacement of Q at that touches exists). In this relaxed setting, it is now ossible that any of the two oints w 1, w 2 lies at infinity, in which case the corresonding disk D 1 or D 2 degenerates into a halflane. This stronger version of Lemma 3.4 is used in the roof of the converse Lemma 3.5, asserting that every edge e in VD (P ) with sufficiently many breakoints has a stable counterart e in VD(P ). LEMMA 3.5. Let, P be a air of oints such that N j [] = for at least three consecutive indices j {0,..., k 1}. Then for each of these indices, excet ossibly for the first and the last one, we have N j [] =. PROOF. Again, assume with no loss of generality that N j [] = for 0 j h 1, with h 3. Suose to the contrary that, for some 1 j h 2, we have N j []. Since N j [] = by assumtion, 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, so that ϕ j 1[, r] < ϕ j 1 [, ] <, which follows because (i) N j 1[] = by assumtion, so ϕ j 1[, ] <, and (ii) ϕ j 1 [, ] ϕ j 1[, ]. See Figure 7 (left). Similarly, we get that either ϕ j 2[, r] < ϕ j 2[, ] < or ϕ j 2[, r] ϕ j 2[, ] = (where the latter can occur only for j = 1). Now alying (the extended version of) Lemma 3.4 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[] =. r b r b u j 1 [] u j [] u j C i [] Figure 7. Left: Proof of Lemma 3.5. If N j [] because some r, lying to the left of the line from to r, satisfies ϕ j [, r] < ϕ j [, ]. Since ϕ j 1 [, ] < ϕ j 1 [, ] <, we have ϕ j 1[, r] < ϕ j 1 [, ]. Right: is j-extremal for. Maintaining an SDG using VD (P). Lemmas 3.4 and 3.5 together imly that SDG can be maintained using the fairly straightforward kinetic algorithm for maintaining the whole VD (P ), rovided by Theorem 3.3. We use VD (P ) to maintain the grah G on P, whose edges are all the airs (, ) P P such that and define an edge e in VD (P ) that contains at least seven breakoints. As shown in Theorem 3.3, this can be done with O(n) storage, O(log n) udate time, and O(k 4 nλ r(n)) udates (for an aroriate r). We claim that G is a (6α, α)-sdg in the Euclidean norm.

7 Indeed, if two oints, P define a 6α-long edge e in VD(P ) then this edge stabs at least six rays u j [] emanating from, and at least six rays u j[] emanating from. Thus, according to Lemma 3.4, VD (P ) contains the edge e with at least four breakoints corresonding to corner lacements of Q at that touch, and at least four breakoints corresonding to corner lacements of Q at that touch. Therefore, e contains at least 8 breakoints, so (, ) G. For the second art, if, P define an edge e in VD (P ) with at least 7 breakoints then, by the interleaving roerty of breakoints, we may assume, without loss of generality, that at least four of these breakoints corresond to P -emty corner lacements of Q at that touch. Thus, Lemma 3.5 imlies that VD(P ) contains the edge e, and that this edge is hit by at least two consecutive rays u j []. But then, as observed in Lemma 2.2, the edge e is α-long in VD(P ). We thus obtain the main result of this section. THEOREM 3.6. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, and let α 0 be a arameter. A (6α, α)-stable Delaunay grah of P can be maintained by a KDS of linear size that rocesses O(nλ r (n)/α 4 ) events, where r is a constant that deends on the degree of motion of P, and that udates the SDG at each event in O(log n) time. 4. A FASTER DATA STRUCTURE The data structure of Theorem 3.6 reuires O(n) storage but the best bound we have on the number of events it may encounter is O (n 2 /α 4 ), which is much larger than the number of events encountered by the data structure of Theorem 2.3. In this section we resent an alternative data structure that reuires O (n/α 2 ) sace and O (n 2 /α 2 ) overall rocessing time. The algorithm is local, and it rocesses each event in O (1/α) time. Notation. We use the directions u i and the associated uantities N i [] and ϕ i [, ] defined in Section 2. We assume that k is even, and write, as in Section 2, k = 2s. We denote by C i the cone (or wedge) with aex at the origin that is bounded by u i and u i+1. Note that C i and C i±s are antiodal. As before, for a vector u, we denote by u[x] the ray emanating from x in direction u. Similarly, for a cone C we denote by C[x] the translation of C that laces its aex at x. Let 0 π/2 be an angle. For a direction u S 1 and for two oints, P, we say that the edge e VD(P ) is -long around the ray u[] if is the Voronoi neighbor of in all directions in the range [u, u+], i.e., for all v [u, u+], the ray v[] intersects e. The -cone around u[] is the cone whose aex is and each of its bounding rays makes an angle of with u[]. j-extremal oints. Let, P, let i be the index such that C i[], and let u j be a direction such that u j, x 0 for all x C i. We say that is j-extremal for if = arg max{, u j C i [] P \ {}}. That is, is the nearest oint to in this cone, in the ( u j )-direction. Clearly, a oint has at most s j-extremal oints, one for every admissible cone C i[], for any fixed j. See Figure 7 (right). For 0 i < k, let C i denote the extended cone that is the union of the seven consecutive cones C i 3,..., C i+3. Let, P, let i be the index such that C i [], and let u j be a direction such that u j, x 0 for all x C i. We say that the oint P is strongly j-extremal for if = arg max{, u j C i[] P \ {}}. We say that a air (, ) P P is (strongly) (j, l)-extremal, for some 0 j, l k 1, if is (strongly) l-extremal for and is (strongly) j-extremal for. C[] h σ + b + 2 a + σ v + v a v e ξ R i,l (w) R w C i+s u i C i B w ξ B i,j (w) ui+1 u l u j Figure 8. Left: Illustration of the setu in Lemma 4.1: the edge e is -long around v[], and the ti" σ + σ of the cone C[] is emty. Right: The oints ξi,l R (w), ξb i,j (w) for an aroriate node w. LEMMA 4.1. Let, P, and let v be a direction such that the edge e aears in VD(P ) and is -long around the ray v[]. Let C[] be the -cone around the ray from through. Then, v, v for all P C[] \ {}. PROOF. Refer to Figure 8 (left). Without loss of generality, we assume that v is the (+x)-direction and that lies above the x- axis and to the right of. (In this case the sloe of the bisector b is negative.) Let v + (res., v ) be the direction that makes a counterclockwise (res., clockwise) angle of with v. Let a + (res., a ) be the intersection of e with v + [] (res., with v []); by assumtion, both oints exist. Let h be the vertical line assing through. Let σ + (res., σ ) be the intersection oint of h with the ray emanating from a + (res., a ) in the direction oosite to v (res., v + ); see Figure 8 (left). Note that a + σ + = 2, and that a + σ + = a + = a +, i.e., a + is the circumcenter of σ +. Therefore σ + = 1 2 σ+ a + =. That is, σ + is the intersection of the uer ray of C[] with h. Similarly, σ is the intersection of the lower ray of C[] with h. Moreover, if there exists a oint x P roerly inside the triangle σ + then a + x < a +, contradicting the fact that a + is on e. So the interior of σ + (including the edges, σ + ) is disjoint from P. Similarly, by a symmetric argument, no oints of P lie inside σ or on its edges, σ. Hence, the ortion of C[] to the right of is oenly disjoint from P, and therefore is a rightmost oint of P (extreme in the v direction) inside C[]. COROLLARY 4.2. Let, P. (i) If the edge e is 3α-long in VD(P ) then there are 0 j, l < k for which (, ) is a (j, l)- extremal air. (ii) If the edge e is 9α-long in VD(P ) then there are 0 j, l < k for which (, ) is a strongly (j, l)-extremal air. PROOF. To rove art (i), choose 0 j, l < k, such that e is α-long around each of u l [] and u j[]. By Lemma 4.1, is u l - extremal in the α-cone C[] around the ray from through. Let i be the index such that C i []. Since the oening angle of C[] is 2α, it follows that C i [] C[], so is l-extremal with resect to, and, symmetrically, is j-extremal with resect to. To rove art (ii) choose 0 j, l < k, such that e is 4α-long around each of u l [] and u j[] and aly Lemma 4.1 as in the roof of art (i). The stable Delaunay grah. We kinetically maintain a (9α, α)- stable Delaunay grah, whose recise definition is given below, using a data-structure which is based on a collection of 2-dimensional orthogonal range trees similar to the ones used in [2]. Fix 0 i < s, and choose a sheared" coordinate frame in which the rays u i and u i+1 form the x- and y-axes, resectively.

8 That is, in this coordinate frame, C i [] if and only if lies in the uer-right uadrant anchored at. We define a 2-dimensional range tree T i consisting of a rimary balanced binary search tree with the oints of P stored at its leaves ordered by their x-coordinates, and of secondary trees, introduced below. Each internal node v of the rimary tree of T i is associated with the canonical subset P v of all oints that are stored at the leaves of the subtree rooted at v. A oint P v is said to be red (res., blue) in P v if it is stored at the subtree rooted at the left (res., right) child of v in T i. For each rimary node v we maintain a secondary balanced binary search tree Ti v, whose leaves store the oints of P v ordered by their y-coordinates. We refer to a node w in a secondary tree Ti v as a secondary node w of T i. Each node w of a secondary tree Ti v is associated with a canonical subset P w P v of oints stored at the leaves of the subtree of Ti v rooted at w. We also associate with w the sets R w P w and B w P w of oints residing in the left (res., right) subtree of w and are red (res., blue) in P v. It is easy to verify that the sum of the sizes of the sets R w and B w over all secondary nodes of T i is O(n log 2 n). For each secondary node w T i and each 0 j < k we maintain the oints ξ R i,j(w) = arg max R w, u j, ξ B i,j(w) = arg max B w, u j, rovided that both R w, B w are not emty. See Figure 8 (right). It is straightforward to show that if (, ) is a (j, l)-extremal air, so that C i[], then there is a secondary node w T i for which = ξ B i,j(w) and = ξ R i,l(w). For each P we define a set N[] containing all oints P for which (, ) is a (j, l)-extremal air, for some air of indices 0 j, l < k. Secifically, for each 0 i < s, and each secondary node w T i such that = ξ R i,l(w) for some 0 l < k, we include in N[] all the oints such that = ξ B i,j(w) for some 0 j < k. Similarly, for each 0 i < s, and each secondary node w T i such that = ξ B i,l(w) for some 0 l < k we include in N[] all the oints such that = ξ R i,j(w) for some 0 j < k. For each 0 i < s, any oint P belongs to O(log 2 n) sets R w and B w, so the size of N[] is bounded by O(s 2 log 2 n). Indeed, may be couled with u to k = 2s neighbors at each of the O(s log 2 n) nodes containing it. For each oint P and 0 l < k we maintain all oints in N[] in a kinetic and dynamic tournament D l [] whose winner minimizes the directional distance ϕ l [, ], as given in (1). That is, the winner in D l [] is N l [] in the Voronoi diagram of {} N[]. We are now ready to define the stable Delaunay grah G that we maintain. For each air of oints, P we add the edge (, ) to G if the following hold. (G1) There is an index 0 l k 1 such that wins the 8 consecutive tournaments D l [],..., D l+7 []. (G2) The oint is strongly (l+3)-extremal and strongly (l+4)- extremal for. In the full version of the aer [1] we rove that G is (9α, α)- stable. We also show how to kinetically maintain (a refined version of) G efficiently, and establish the following theorem. THEOREM 4.3. Let P be a set of n moving oints in R 2 under algebraic motion of bounded degree, and let α > 0 be a arameter. A (9α, α)-sdg of P can be maintained using a data structure that reuires O((n/α 2 ) log n) sace and encounters two tyes of events: swa events and tournament events. There are O(n 2 /α) swa events, each rocessed in O(log 4 (n)/α) time. There are O((n/α) 2 r+2 (log(n)/α) log 2 n log(log(n)/α)) tournament events, which are handled in a total of O((n/α) 2 r+2 (log(n)/α) log 2 n log 2 (log(n)/α)) rocessing time. The worst-case rocessing time of a tournament event is O(log 2 (log(n)/α)). Remark: Comaring this algorithm with the sace-inefficient one of Section 2, we note that they both use the same kind of tournaments, but here much fewer airs of oints (O (n/α 2 ) instead of O(n 2 /α)) articiate in the tournaments. The rice we have to ay is that the test for an edge to be stable is more involved. Moreover, keeing track of the subset of airs that articiate in the tournaments reuires additional work, which is facilitated by the range trees T i. 5. PROPERTIES OF SDG We conclude the aer by establishing some of the roerties of stable Delaunay grahs. Near co-circularities do not show u in an SDG. Consider a critical event during the kinetic maintenance of the full Delaunay triangulation, in which four oints a, b, c, d become co-circular, in this order along their circumcircle, with this circle being emty. Just before the critical event, the Delaunay triangulation involved two triangles, say, abc, acd. The Voronoi edge e ac shrinks to a oint (namely, to the circumcenter of abcd at the critical event), and, after the critical co-circularity, is relaced by the Voronoi edge e bd, which exands from the circumcenter as time rogresses. Our algorithm will detect the ossibility of such an event before the criticality occurs, when e ac becomes α-short (or even before this haens). It will then remove this edge from the stable subgrah, so the actual co-circularity will not be recorded. The new edge e bd will then be detected by the algorithm only when it becomes sufficiently long (if at all), and will then enter the stable Delaunay grah. In short, critical co-circularities do not arise at all in our scheme. As noted in the introduction, a Delaunay edge ab is just about to become α-short or α-long when the sum of the oosite angles in its two adjacent Delaunay triangles is π α (see Figure 1 (left)). This shows that changes in the stable Delaunay grah occur when the co-circularity defect of a nearly co-circular uadrule (i.e., the difference between π and the sum of oosite angles in the uadrilateral sanned by the uadrule) is between α and cα, where c is the constant used in our definitions in Section 3 or Section 4. Note that a degenerate case of co-circularity is a collinearity on the convex hull. Such collinearities also do not show u in the stable Delaunay grah. A hull collinearity between three nodes a, b, c is detected before it haens, when (or before) the corresonding Voronoi edge becomes α-short, in which case the angle acb, where c is the middle oint the (near-)collinearity becomes π α (see Figure 9 (left)). Therefore a hull edge is removed if the Delaunay triangle is almost collinear. The edge re-aears when its corresonding Voronoi edge is long enough, as before. SDGs are not too sarse. Consider the Voronoi cell Vor() of a oint, and suose that has only one α-long edge e. Since the angle at which sees e is at most π, the sum of the angles at which sees the other edges is at least π, so Vor() has at least π/α α-short edges. Let m 1 denote the number of oints with this roerty. Then the sum of their degrees in DT(P ) is at least m 1 (π/α + 1). Similarly, if m 0 oints do not have any α-long