D.T. Lee, "On k-nearest neighbor Voronoi diagrams in the plane," IEEETrans. Comput. 31 (1982) 478--487.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with very different methods used in the solutions. Problem 3 can be solved by constructing the order (k 1) Voronoi diagram, and then for each point p determining the other k points lying in the same Voronoi region. For small values of k in the plane, this is a fairly efficient method. Lee [16] showed how to construct the order k Voronoi diagram in O(k 2 n log n) time, and Aggarwal et al. 2] have since improved that result to O(k 2 n n log n) time. Dickerson et al. 11] presented an asymptotically faster algorithm requiring O(n log n kn log k) time for the planer case; as ....

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

.... element responsible for shorts within reg P (s) Because s is the element inducing the critical radius for every point x 2 reg P (s) we drop P and say that s is the owner of reg P (s) This 2nd order Voronoi diagram of polygons is based on the concept of kth order Voronoi diagram of line segments [9] which is defined as a planar subdivision such that each region is closest to k line segments. As described in [9] we can visualize the kth order Voronoi diagram as a finer subdivision of the cells in the (k Gamma 1) order diagram, where the first order diagram is the ordinary Voronoi diagram. ....

....x 2 reg P (s) we drop P and say that s is the owner of reg P (s) This 2nd order Voronoi diagram of polygons is based on the concept of kth order Voronoi diagram of line segments [9] which is defined as a planar subdivision such that each region is closest to k line segments. As described in [9], we can visualize the kth order Voronoi diagram as a finer subdivision of the cells in the (k Gamma 1) order diagram, where the first order diagram is the ordinary Voronoi diagram. The size of the 2nd order Voronoi diagram cannot be more than twice the size of ordinary Voronoi diagram i.e. it ....

D. T. Lee, "On k-nearest neighbor Voronoi diagrams in the plane," IEEE Trans. Comput., Vol. C-31, No. 6, June 1982, 478-487.

....that s is the owner of regP (s) The size of the 2nd order Voronoi diagram cannot be more than twice the size of ordinary Voronoi diagram i.e. it is linear in the size of the input. This 2nd order Voronoi diagram of polygons is based on the concept of kth order Voronoi diagram of line segments [8] which is defined as a planar subdivision such that each region is closest to k line segments. To derive the 2nd order diagram within the Voronoi cell of each polygon P we only need to consider the Voronoi neighbors of reg(P ) In fact, as shown in Figure 6, the 2nd 0 e V R Figure 7. 2nd order ....

D. T. Lee, "On k-nearest neighbor Voronoi diagrams in the plane," IEEE Trans. Comput., Vol. C-31, No. 6, June 1982, 478-487.

....in the plane, AIKS91] contains an O(p 2:5 n log p n log n) algorithm for the MDP problem and an O(p 2 n log n) algorithm for the MVP 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 [Le82, PS85]. Other work has addressed placement problems where the objective functions are different from the above. For example, the traditional facility location problems are concerned with minimizing the maximum distance from a node to a nearest facility (p center problem) or minimizing the sum of the ....

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

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