Delaunay Triangulation: Incremental Construction

Transcription

1 Chapter 6 Delaunay Triangulation: Incremental Contruction In the lat lecture, we have learned about the Lawon ip algorithm that compute a Delaunay triangulation of a given n-point et P R 2 with O(n 2 ) Lawon ip. One can actually implement thi algorithm to run in O(n 2 ) time, and there are point et where it may take Ω(n 2 ) ip. In thi lecture, we will dicu a dierent algorithm. The nal goal i to how that thi algorithm can be implemented to run in O(n log n) time; thi lecture, however, i concerned only with the correctne of the algorithm. Throughout the lecture we aume that P i in general poition (no 3 point on a line, no 4 point on a common circle), o that the Delaunay triangulation i unique (Corollary 5.17). There are technique to deal with non-general poition, but we don't dicu them here. 6.1 Incremental contruction The idea i to build the Delaunay triangulation of P by inerting one point after another. We alway maintain the Delaunay triangulation of the point et R inerted o far, and when the next point come along, we imply update the triangulation to the Delaunay triangulation of R {}. Let DT(R) denote the Delaunay triangulation of R P. To avoid pecial cae, we enhance the point et P with three articial point far out. The convex hull of the reulting point et i a triangle; later, we can imply remove the extra point and their incident edge to obtain DT(P). The incremental algorithm tart o with the Delaunay triangulation of the three articial point which conit of one big triangle encloing all other point. (In our gure, we uppre the far-away point, ince they are merely a technicality.) Now aume that we have already built DT(R), and we next inert P \ R. Here i the outline of the update tep. 1. Find the triangle = (p, q, r) of DT(R) that contain, and replace it with the three triangle reulting from connecting with all three vertice p, q, r; ee Figure 88

2 Geometry: C&A Incremental contruction Figure 6.1: Inerting into DT(R): Step We now have a triangulation T of R {}. 2. Perform Lawon ip on T until DT(R {}) i obtained; ee Figure 6.2 Figure 6.2: Inerting into DT(R): Step 2 How to organize the Lawon ip. The Lawon ip can be organized quite ytematically, ince we alway know the candidate for bad edge that may till have to be ipped. Initially (after tep 1), only the three edge of can be bad, ince thee are the only edge for which an incident triangle ha changed (by inerting in Step 1). Each of the three new edge i good, ince the 4 vertice of it two incident triangle are not in convex poition. Now we have the following invariant (part (a) certainly hold in the rt ip): 89

3 Chapter 6. Delaunay Triangulation: Incremental ContructionGeometry: C&A 2015 (a) In every ip, the convex quadrilateral Q in which the ip happen ha exactly two edge incident to, and the ip generate a new edge incident to. (b) Only the two edge of Q that are not incident to can become bad through the ip. We will prove part (b) in the next lemma. The invariant then follow ince (b) entail (a) in the next ip. Thi mean that we can maintain a queue of potentially bad edge that we proce in turn. A good edge will imply be removed from the queue, and a bad edge will be ipped and replaced according to (b) with two new edge in the queue. In thi way, we never ip edge incident to ; the next lemma prove that thi i correct and at the ame time etablihe part (b) of the invariant. Lemma 6.1 Every edge incident to that i created during the update i an edge of the Delaunay graph of R {} and thu an edge that will be in DT(R {}). It eaily follow that edge incident to will never become bad during the update tep. 1 Proof. Let u conider one of the rt three new edge, p, ay. Since the triangle ha a circumcircle C trictly containing only ( i in DT(R)), we can hrink that circumcircle to a circle C through and p with no interior point, ee Figure 6.3 (a). Thi prove that p i in the Delaunay graph. If t i an edge created by a ip, a imilar argument work. The ip detroy exactly one triangle of DT(R). It circumcircle C contain only, and hrinking it yield an empty circle C through and t. Thu, t i in the Delaunay graph alo in thi cae. q C p r C (a) New edge p incident to created in Step 1 C t C (b) New edge t incident to created in Step 2 Figure 6.3: Newly created edge incident to are in the Delaunay graph 1 If uch an edge wa bad, it could be ipped, but then it would be gone forever according to the lifting map interpretation from the previou lecture. 90

4 Geometry: C&A The Hitory Graph Figure 6.4: The hitory graph: one triangle get replaced by three triangle 6.2 The Hitory Graph What can we ay about the performance of the incremental contruction? Not much yet. Firt of all, we did not pecify how we nd the triangle of DT(R) that contain the point to be inerted. Doing thi in the obviou way (checking all triangle) i not good, ince already the nd tep would then amount to O(n 2 ) work throughout the whole algorithm. Here i a marter method, baed on the hitory graph. Denition 6.2 Given R P (regarded a a equence that reect the inertion order), the hitory graph of R i a directed acyclic graph whoe vertice are all triangle that have ever been created during the incremental contruction of DT(R). There i a directed edge from to whenever ha been detroyed during an inertion tep, ha been created during the ame inertion tep, and overlap with in it interior. It follow that the hitory graph contain triangle of outdegree 3, 2 and 0. The one of outdegree 0 are clearly the triangle of DT(R). The triangle of outdegree 3 are the one that have been detroyed during Step 1 of an inertion. For each uch triangle, it three outneighbor are the three new triangle that have replaced it, ee Figure 6.4. The triangle of outdegree 2 are the one that have been detroyed during Step 2 of an inertion. For each uch triangle, it two outneighbor are the two new triangle created during the ip that ha detroyed, ee Figure 6.5. The hitory graph can be built during the incremental contruction at aymptotically no extra cot; but it may need extra pace ince it keep all triangle ever created. Given the hitory graph, we can earch for the triangle of DT(R) that contain, a follow. We tart from the big triangle panned by the three far-away point; thi one certainly 91

5 Chapter 6. Delaunay Triangulation: Incremental ContructionGeometry: C&A 2015 Figure 6.5: The hitory graph: two triangle get replaced by two triangle contain. Then we follow a directed path in the hitory graph. If the current triangle till ha outneighbor, we nd the unique outneighbor containing and continue the earch with thi neighbor. If the current triangle ha no outneighbor anymore, it i in DT(R) and contain we are done. Type of triangle in the hitory graph. After each inertion of a point, everal triangle are created and added to the hitory graph. It i important to note that thee triangle come in two type: Some of them are valid Delaunay triangle of R {}, and they urvive to the next tage of the incremental contruction. Other triangle are immediately detroyed by ubequent Lawon ip, becaue they are not Delaunay triangle of R {}. Thee ephemeral" triangle will give u ome headache (though not much) in the algorithm' analyi in the next chapter. Note that, whenever a Lawon ip i performed, of the two triangle detroyed one of them i alway a valid" triangle from a previou iteration, and the other one i an ephemeral" triangle that wa created at thi iteration. The ephemeral triangle i alway the one that ha, the newly inerted point, a a vertex. 6.3 The tructural change Concerning the actual update (Step 1 and 2), we can make the following Obervation 6.3 Given DT(R) and the triangle of DT(R) that contain, we can build DT(R {}) in time proportional to the degree of in DT(R {}), which i the number of triangle of DT(R {}) containing. Indeed, ince every ip generate exactly one new triangle incident to, the number of ip i the degree of minu three. Step 1 of the update take contant time, and 92

6 Geometry: C&A The tructural change ince alo every ip can be implemented in contant time, the obervation follow. In the next lecture, we will how that a clever inertion order guarantee that the earch path travered in the hitory graph are hort, and that the tructural change (the number of new triangle) i mall. Thi will then give u the O(n log n) algorithm. Exercie 6.4 For a equence of n pairwie ditinct number y 1,..., y n conider the equence of pair (min{y 1,..., y i }, max{y 1,..., y i }) i=0,1,...,n (min := +, max := ). How often do thee pair change in expectation if the equence i permuted randomly, each permutation appearing with the ame probability? Determine the expected value. Quetion 26. Decribe the algorithm for the incremental contruction of DT(P): how do we nd the triangle containing the point to be inerted into DT(R)? How do we tranform DT(R) into DT(R {})? How many tep doe the latter tranformation take, in term of DT(R {})? 27. What are the two type of triangle that the hitory graph contain? 93