Lee, D.T. -- On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31: 478-487, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....plane. The poset consists of all sets of labels that correspond with a subset of sites that defines some non empty Voronoi region in some k th order Voronoi diagram. Higher order Voronoi diagrams have been investigated by numerous people. Many results are published in an article by D.T. Lee, see [Le]. A survey is given in Edelsbrunners book on algorithms in combinatorial geometry, see [Ed] He ends his paragraph on the complexity of higher order Voronoi diagrams with the following remark. Interestingly enough, the number of regions of V k (S) is thus exactly kn Gamma k 1 if n is odd, ....

....of S as already the ordinary Voronoi diagram shows. But the following theorem gives expressions for those numbers, depending on n, k and numbers of unbounded regions. Let f k denote the number of unbounded regions in the k th order Voronoi diagram. By definition f 0 : 0. Theorem 3. 3 [Ed, Le] Let S be in general position. Then the number of vertices, edges and regions in the k th order Voronoi diagram can be expressed as follows. i) v k = 2(f k Gamma 1) Gamma f 5 (ii) e k = 3(f k Gamma 1) Gamma f (iii) f k = 2k Gamma 1)n Gamma (k Gamma 1) Gamma P k ....

Lee, D.T. -- On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31: 478-487, 1982.

....To summarize, a close type vertex of a higher order Voronoi diagram remains in d successive diagrams. More generally, a h face remains in d Gamma h successive diagrams. Complexity results The order k Voronoi diagram has been introduced in [SH75] to deal with k closest points problems. Lee [Lee82] gives the following result : Property 1.3 In the plane, the size of the order k Voronoi diagram is O(k(n Gamma k) The size of orders k Voronoi diagrams is thus O(nk ) The random sampling technique (Section 2.3.1) shows the following worst case result in dimension d [CS89] Property ....

....a bound for jF j (S)j, more precise that the general bound given by Theorem 2.3 for jFj (S)j. Lemma 5.12 The number of triangles having width j is jF j (S)j (2j 1)n Proof. jF j (S)j is exactly the number of close type vertices of the order j 1 Voronoi diagram. This number is computed in [Lee82] which gives the result. 2 We propose here an alternative proof for Lemma 4.10, owing to the preceding lemma. jG j (S)j = O(n(j 1) Proof. We first notice that a given segment joining two points of S is an edge of at most 2j 2 triangles of width less than j : in Figure 5.6, every ....

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, C-31:478--487, 1982.

....Triangle area: A(v; p; q) A(v; p; q) 5. Di erence between distances: D(v; p; q) jd(v; p) d(v; q)j. We show in the video that the rst two functions generate very di erently looking surfaces, but almost identical Voronoi diagrams. Both are equivalent to the second order Voronoi diagram [2]. The rst and third generate the same shaped surfaces, but because they are at di erent heights above the xy plane, the Voronoi diagrams are quite di erent. Consider, for example, the Voronoi diagram of three points with respect to the 2 site triangle area distance function. 6 3 3 p x r q ....

D.T. Lee, On k-nearest neighbor Voronoi diagrams in the plane, IEEE Trans. on Computing, 31 (1982), 478-487.

....edge e ij with a perturbation of minimum size, the circle centered at c with radius r ffi ij cannot contain any other point and thus the region of V 4 (S) corresponding to the points p i , p j , p k and p l is nonempty. Because V 4 (S) has linear complexity and can be computed in time O(n log n) [16], we conclude observing that in order to compute the tolerance of all the edges of DT (S) we can explore all the regions of V 4 (S) and, for each region, compute in O(1) the corresponding annulus. It is worth noticing that, if DT (S) is known and its degree is bounded by a constant, the ....

D. T. Lee, On k-nearest neighbor Voronoi diagram in the plane, IEEE Trans. Comput. 31 (1977) 478-487.

....in the d space, 1 i n, partitions the space into regions such that each point within a fixed region has the same i closest sites. The regions of V D i (S) are convex polyhedra. The number of non empty regions in the two dimensional ith order Voronoi diagram can be bounded by O(i(n Gamma i) [53]. Bentley and Maurer [14] presented a technique which extracts the sites of the Voronoi region containing q in the ith order Voronoi diagram of S for i = 2 0 log n; 2 1 log n; 2 2 log n; consecutively. It stops, if one of the i sites that are assigned to the Voronoi cell containing q is ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, C-31:478--487, 1982.

.... 2 (2 n ) be a set of distinct points, then the farthest point Voronoi polygon, i fp p v defined as: i fp p v = i j j i p P p p p d p p d p ) i j j p p p p p p d p p d p , max ) The FVD is, in fact, an order k Voronoi diagram (Lee, 1982). If i p is the farthest point from p , then the set of points P i p is the set of the first, the second, the (n 1) th nearest point from i p . Thus the ) i fp p v is the same as the order (n 1) Voronoi polygon associated with P i p from the order (n 1) Voronoi diagram ....

Lee, D. T. (1982). On k-nearest neighbor Voronoi diagrams in the plane, IEEE Transactions on Computers, C31: 478-487.

....space complexity. We will omit the computations of these constants in this paper. 9 Remark 3.1 For k 2, we can trivially deduce the following bounds : s 0 (n) s 1 (n) s 2 (n) O i 2 b d 2 c 1 n d d 2 e j and similarly for bicycles. Remark 3. 2 In the two dimensional case, from [Lee82] s k (n) and b k (n) are O(k(n Gamma k) The sequel analyzes the insertion of the last site M of the set M of the n already inserted elements. Analysis of the randomized expected cost of Procedure location We compute the expected number of nodes (n) visited to locate the n th site M . A ....

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, C-31:478--487, 1982.

.... V k (S) for a set S of n point sites has O(k(n Gamma k) vertices, edges, and faces, and can be obtained from the order k Gamma 1 implicit Voronoi diagram V k Gamma1 (S) by intersecting each face of V k Gamma1 (S) with the (order 1) Voronoi diagram of a suitable subset of the vertices of S [42]. As shown in [42, 15] V k (S) can be computed in O(k(n Gamma k) p n log n) time. Since the construction is based on iteratively computing Voronoi diagrams by using the incircle test, which is the most expensive operation in terms of degree, the overall degree of the preprocessing is 4 (Lemma ....

.... a set S of n point sites has O(k(n Gamma k) vertices, edges, and faces, and can be obtained from the order k Gamma 1 implicit Voronoi diagram V k Gamma1 (S) by intersecting each face of V k Gamma1 (S) with the (order 1) Voronoi diagram of a suitable subset of the vertices of S [42] As shown in [42, 15], V k (S) can be computed in O(k(n Gamma k) p n log n) time. Since the construction is based on iteratively computing Voronoi diagrams by using the incircle test, which is the most expensive operation in terms of degree, the overall degree of the preprocessing is 4 (Lemma 5) Hence, we obtain. ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--487, 1982.

.... 2 V k (H k ; S) d T 2( x; p) T 1 d T 2( x; q) 8p 2 H k ; 8q 2 S n H k ; which is equivalent to x 2 V k (H k ; S) d T 2( x; q) T 1 d T 2( x; p) 8q 2 S n H k ; 8p 2 H k ; or x 2 V k (H k ; S) x 2 V n k (S n H k ; S) The following lemma, whose Euclidian version can be found in [4], describes the proper points of T 2 which lie on edges of some order k Voronoi diagram. For a description of points at in nity, see Lemma 3 and Theorem 4. 4 P. J. Rezende and R. B. Westrupp Lemma 2: Let V k (S) be the order k Voronoi diagram of a set S of sites in T 2 . p = 2 is a point of ....

....of a set S of sites in T 2 . p = 2 is a point of an edge B k (s i ; s j ) of V k (S) if and only if the circle centered at p with radius d T 2(p; s i ) d T 2(q; s j ) contains k 1 sites of S in its interior. Proof: It is a simple generalization of the proof of lemma 3 due to Lee [4]. Theorem 2: Let v, v 2 T 2 be the proper circumcenters of s a , s b , s c 2 S, with s a , s b , s c on the same range of T 2 . Let H denote the set of sites that are closer to v than s a , s b , s c are: H = fz 2 S : d T 2(v; z) T 1 d T 2(v; s a )g and let k = jHj. Then, v is a ....

[Article contains additional citation context not shown here]

LEE, D. T. On k-Nearest Neighbor Voronoi Diagrams in the Plane. IEEE Transactions on Computers, c{31, n. 6, June 1982.

.... 91] contains an O(p 2:5 n log p n log n) algorithm for the (Min Dia) problem and an O(p 2 n log n) algorithm for the (Min Var) problem, and it is observed that these algorithms extend to higher dimensions. These algorithms are based on the construction of p th order Voronoi diagrams [Lee82, PS85] Problems involving the placement of p facilities so as to minimize suitable cost measures have been studied extensively in the literature (see [BE95, AI 91, DL 93, KP93, RKM 93, EE94] and the references cited therein) These problems can roughly be divided into two main ....

D. T. Lee, On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C--31:478--487, 1982.

....higher dimensions, the current upper bound is only slightly better than the O(n bd=2c k dd=2e ) bound for the larger ( k) level. There is a case where the exact complexity of the k level is known: if d = 3 and all input planes are nonredundant, then the k level has size Theta(nk) for k n=2 [32, 46]. The k level in this case is related to the order k Voronoi diagram of n points in the Euclidean plane a natural extension to one of the most fundamental and useful geometric structures, the Voronoi diagram. 3 Some of the algorithmic results obtained by Agarwal et al. 2] on the k level can ....

....of Agarwal and Matousek [6] with the known combinatorial bound [35] Hence, Theorem 3.3 implies: Corollary 3. 4 The k level in an arrangement of n lines in IR 2 can be constructed in expected time O(n log n nk 1=3 log 2=3 k) 10 year time bound references 1982 O(nk 2 log n) Lee [46] 1986 O(n 3 ) Edelsbrunner, O Rourke, and Seidel [39] 1986 O(nk p n log n) Edelsbrunner [36] 1987 O(n 2 nk log 2 n) Chazelle and Edelsbrunner [26] 1987 O(n 1 k) rand. Clarkson [30] 1989 O(n log n nk 2 ) Aggarwal, Guibas, Saxe, and Shor [9] 1991 O(n log n nk 2 ) rand. ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--287, 1982.

....are in general much more complex curves. The reason for this is that the Voronoi diagram contains only portions of bisectors of pairs which share one site, that is, of the form (p; q) and (p; r) The combinatorial complexity of the second order (Euclidean) Voronoi diagram is known to be Theta(n) [PS,Le]. The diagram can be computed in optimal Theta(n log n) time and Theta(n) space. Similarly, the diagrams V (f) S (S) and V (f) M (S) are identical to the second order furthest neighbor Voronoi diagram of S with respect to the usual Euclidean distance function. The bounds on the complexity ....

D.T. Lee, On k-nearest neighbor Voronoi diagrams in the plane, IEEE Trans. on Computing, 31 (1982), 478--487.

....denotes the number of k level vertices lying on h. Repeating this process for every h 2 H and gluing the structure together yield an overall running time O 0 X h2H (n log n m h log 1 n) 1 A = O(n 2 log n m log 1 n) where m, the size of the output diagram, is known to be O(nk) [31]. 2 Using more complicated machinery (namely, shallow cuttings [11] we can make the above result sensitive to k, leading to the current best deterministic time bound for planar order k Voronoi diagrams: O(nk log 1 k Delta (log n= log k) O(1) However, simpler and slightly faster ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--287, 1982.

....c j et une place m emoire O i k d d 1 2 e n b d 1 2 c j . L algorithme est simple et des r esultats exp erimentaux sont pr esent es. 1 Introduction The order k Voronoi diagram has been introduced in [3] in order to deal with k closest points and related distance relationships. Lee [4] gave the first algorithm for constructing the order k Voronoi diagram of a set of n points (or sites) in the plane. This algorithm constructs the order k Voronoi diagram from the order (k Gamma 1) Voronoi diagram in time O(kn log n) Thus the order k Voronoi diagram (in fact, the family of all ....

....2 The following lemma is a direct consequence of the bound on the size of higher order Voronoi diagrams. Lemma 3.3 The number of triangles having width j is jT j j (2j 1)n Proof : jT j j is exactly the number of close type vertices of the order j 1 Voronoi diagram. The result follows from [4]. 2 The next lemma is given in [10] We propose an alternative proof here. Lemma 3.4 The number of bicycles having width at most j is jC j j = O(n(j 1) 3 ) Proof : We first notice that a given segment joining two points of S is an edge of at most 2j 2 triangles of width less than j. We ....

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, C-31:478--487, 1982.

....higher dimensions, the current upper bound is only slightly better than the O(n bd=2c k dd=2e ) bound for the larger ( k) level. There is a case where the exact complexity of the k level is known: if d = 3 and all input planes are nonredundant, then the k level has size Theta(nk) for k n=2 [29, 41]. The k level in this case is related to the order k Voronoi diagram of n points in the Euclidean plane a natural extension to one of the most fundamental and useful geometric structures, the Voronoi diagram. Some of the algorithmic results obtained by Agarwal et al. 2] on the k level can be ....

....of Agarwal and Matousek [6] with the known combinatorial bound [32] Hence, Theorem 3.3 implies: Corollary 3. 4 The k level in an arrangement of n lines in IR 2 can be constructed in expected time O(n log n nk 1=3 log 2=3 k) year time bound references 1982 O(nk 2 log n) Lee [41] 1986 O(n 3 ) Edelsbrunner, O Rourke, and Seidel [35] 1986 O(nk p n log n) Edelsbrunner [33] 1987 O(n 2 nk log 2 n) Chazelle and Edelsbrunner [23] 1987 O(n 1 k) rand. Clarkson [27] 1989 O(n log n nk 2 ) Aggarwal, Guibas, Saxe, and Shor [9] 1991 O(n log n nk 2 ) rand. ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--287, 1982.

....plane. The poset consists of all sets of labels that correspond with a subset of sites that defines some non empty Voronoi region in some k th order Voronoi diagram. Higher order Voronoi diagrams have been investigated by numerous people. Many results are published in an article by D.T. Lee, see [Le]. A survey is given in Edelsbrunners book on algorithms in combinatorial geometry, see [Ed] He ends his paragraph on the complexity of higher order Voronoi diagrams with the following remark. Interestingly enough, the number of regions of V k (S) is thus exactly kn Gamma k 2 1 if n is odd, ....

....of S as already the ordinary Voronoi diagram shows. But the following theorem gives expressions for those numbers, depending on n, k and numbers of unbounded regions. Let f 1 k denote the number of unbounded regions in the k th order Voronoi diagram. By definition f 1 0 : 0. Theorem 3. 3 [Ed, Le] Let S be in general position. Then the number of vertices, edges and regions in the k th order Voronoi diagram can be expressed as follows. i) v k = 2(f k Gamma 1) Gamma f 1 k (ii) e k = 3(f k Gamma 1) Gamma f 1 k (iii) f k = 2k Gamma 1)n Gamma (k 2 Gamma 1) Gamma P k i=1 ....

Lee, D.T. -- On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31: 478-487, 1982.

....foothold inside V i ) The union W of the W i for all the cells V i of the diagram is the whole set of l feasible configurations. The sizes of the order l Gamma 1 and l Voronoi diagrams, as well as the size of their superposition are O(nl) and they can be computed in O(n log n nl 2 ) time. Lee82, AGSS87] Constructing W can be done within the same time bound. Unfortunatly, the Voronoi diagram does not solve the problem of the stability of the robot. It may happen that the nearest footholds placement is not stable while there exists another placement which is both feasible and stable. The ....

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--487, 1982.

....is used to store the Voronoi vertices and to enable the algorithm to avoid duplication. turn might be useful for constructing approximations to polyhedra with the smallest number of triangles (see [20] and [8] Another application is robotic path planning [13] 14] The order k Voronoi diagram [11] [21] can be used to rapidly determine the k nearest sites (vertices, faces, or edges) to a point inside a polyhedron. This can be useful to a physical simulation of the properties of a potential function based on the boundary of the polyhedron. At present, there is no efficient 2 exact ....

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computing C31: 478-487, 1982.

....1. Both bounds are attained for sets in convex position. Recall that in the plane a tight bound of E k kn and G j (j 1)n is known for k n=2 and j n=2 Gamma 1 ( AG, Pe] In IR , the number of j facets is 2(j 1) n Gamma j Gamma 2) for every set of n points in convex position ([Lee, CS, Sh]) and by Theorem 3 below this implies that the number of k sets is n 2(n Gamma k Gamma 1) k Gamma 1) see also [GH] The theorem above quotes the resulting numbers of ( j) facets and ( k) sets for point sets in convex position. In our proof we will show that we can always move a point set ....

D. T. Lee, On k-nearest neighbor Voronoi diagrams in the plane, IEEE Trans Comput C-31 (1982) 478-- 487

No context found.

D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478487, 1982.

No context found.

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, 31(6):478--487, 1982.

No context found.

D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Transactions on Computers, C-31:478--487, 1982.

No context found.

Lee, D. T. #1982# On k-nearest neighbor Voronoi diagrams in the plane, IEEE Trans. Comput. C-31,

No context found.

D T Lee, On k-nearest neighbor Voronoi diagram in the plane, IEEE Trans. Comput. 31 (1982) 478-487.

No context found.

D.T. Lee. On k--nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C--31:478--487, 1982.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC