K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R of order 1 to k. In Proc. 22nd ACM Sympos. Theory Comput., pages 322--330, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 (H) I k (H) by first computing the minimum of I 0 (H) by LP) then repeatedly removing a solution s defining halfspace and reoptimizing, to generate I 1 (H) minima from the I 0 (H) minimum, I 2 (H) minima from the I 1 (H) minima, and so on. As the ( k) level has O(k ) local minima [43], the cost of the algorithm is dominated by the cost of O(k ) dynamic LP operations. The general 2 d problem can be lifted to a feasible 3 d problem, and with the appropriate data structures [45] these O(k ) operations can be carried out in O(n log n k n) time. The first term has ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R of order 1 to k. In Proc. 22nd ACM Sympos. Theory Comput., pages 322--330, 1990.

....diagram algorithm is faster. Finally,fork= Omega 1; the fastest algorithm is based on another Voronoi algorithm of Chazelle and Edelsbrunner [6] and runs in time O(n n) Mulmuley describes an algorithm that constructs the kth order Voronoi diagram 2 e b 2 c log n k )[30]. To find minimum variance sets in higher dimensions, we use Mulmuley s algorithm as a subroutine within eachneighbor set. We improve the previous time bound of O(n ) 2] v(d) 1 n log k) and space O(n k v(d) where v(d) if d is even, and if d is odd. 6 Dynamization ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R of order 1 to k.In22nd ACM Symp. Theory Comput., pages 322--330, 1990.

....IR 3 can be constructed in O(n log n nk 2 ) expected time. Remarks : 1. Although our algorithm is optimal for worst case output, the running time is not the best possible if the ( k) level has size smaller than Theta(nk 2 ) It remains open to devise an optimal output sensitive algorithm [6, 18, 51]. 9 2. Derandomization seems to require modification to our algorithm because the sample size is quite large. Appendix A.2 sketches an optimal deterministic method for ( k) levels for large k (specifically, log k = Omega Gamma191 n) This method works in any fixed dimension but uses advanced ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In Proc. 22nd ACM Sympos. Theory Comput., pages 322--330, 1990.

....IR 3 can be constructed in O(n log n nk 2 ) expected time. Remarks: 1. Although our algorithm is optimal for worst case output, the running time is not the best possible if the ( k) level has size smaller than Theta(nk 2 ) It remains open to devise an optimal output sensitive algorithm [6, 17, 46]. 2. Derandomization seems to require modification to our algorithm because the sample size is quite large. Appendix A.2 sketches an optimal deterministic method for ( k) levels for large k (specifically, log k = Omega Gamma191 n) This method works in any fixed dimension but uses advanced ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In Proc. 22nd ACM Sympos. Theory Comput., pages 322--330, 1990.

....Chapter 4. Higher Dimensional Convex Hulls 78 problems [ERvK93] or to the smallest k enclosing circle problem; see Matousek s paper [Mat94] As we now show, Matousek s approach on linear programming with violations can in fact be used to improve an output sensitive algorithm by Mulmuley [Mul90] for constructing ( k) levels i.e. the i level for all i = 0; 1; k of a non redundant arrangement A(H) of n hyperplanes in E d . Here, A(H) is non redundant if for every h 2 H, the upper envelope of H Gamma fhg coincides with the upper envelope of H. Mulmuley s algorithm can be ....

....n) Here, we show how to reduce the first term of the running time of Mulmuley s algorithm to O(n 2 Gamma2= bd=2c 1) k d Gamma1 ) Using the lifting map from Section 3. 3, this result can be used in the construction of Voronoi diagrams of order 0; 1; k in one dimension lower; see [ES86, Mul90]. Theorem 4.5.3 We can compute i levels in a non redundant arrangement A(H) of n hyperplanes in E d for all i = 0; 1; k in O(n 2 Gamma2= bd=2c 1) k d Gamma1 f log n) time, where f is the output size. Proof: Let L i (H) denote the boundary of the i level in A(H) and let f i be ....

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K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In Proceedings of the 22rd Annual ACM Symposium on Theory of Computing, pages 322--330, 1990.

.... algorithm is based on another Voronoi algorithm of Chazelle and Edelsbrunner [6] and runs in time O(n 2 log 2 n) Mulmuley describes an algorithm that constructs the kth order Voronoi diagram of a set of n points in IR d , in time O(k d d 1 2 e n b d 1 2 c log n k d n 2 ) [30]. To find minimum variance sets in higher dimensions, we use Mulmuley s algorithm as a subroutine within each neighbor set. We improve the previous time bound of O(n d 1 ) 2] Theorem 5.8. We can find the minimum variance k point subset of a set of n points in IR d , in time O(k (d 1) 2 n ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In 22nd ACM Symp. Theory Comput., pages 322--330, 1990.

....in these bounds can be reduced to O(n log k) in two dimensions or to O(n log log n n log k) in higher dimensions; if randomization is allowed, this O(n log log n) term can even be removed. As an aside, we point out that the Matousek s results [25] can be used to improve an algorithm by Mulmuley [30] for constructing ( k) levels of a nonredundant arrangement of n hyperplanes in E d . The algorithm is an extension of Seidel s output sensitive convex hull algorithm [41] and runs in O(n 2 k d Gamma1 f log n) time for an f face output. We decrease the time bound to O(n ....

....and Widmayer [40] observed. The techniques here may also be applicable to the infeasible case of linear programming with k violated constraints, or to the smallest k enclosing circle problem; see Matousek s paper [25] Finally, we mention an improvement to Mulmuley s output sensitive algorithm [30] for constructing ( k) levels. The algorithm assumes that the input hyperplanes H are nonredundant , i.e. every hyperplane in H supports the upper envelope of H . For applications to ( k) order Voronoi diagrams, this assumption is automatically satisfied. Theorem 7.3 We can compute i levels in ....

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K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In Proc. 22nd ACM Sympos. Theory of Comput., 322--330, 1990.

....i level. Note that the 0 minimum of H is the solution x (H) of the linear programming problem minfx 1 j x 2 P(H)g. In addition to bounding the number of (k) minima, Mulmuley showed some bounds on related quantities, and conjectured that the number of i minima is O(i d Gamma1 ) for every i [7]. This conjecture is confirmed here by the bound Gamma i d Gamma1 d Gamma1 Delta , proven in the next section using the same technique as for bounds on (k) sets and (k) minima. This i minima bound is of course not new for i = 0 and i = 1, and it isn t even new for i = n=2: using (projective ....

K. Mulmuley. Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k. In Proc. 22nd Annual SIGACT Symp., pages 322--330, 1990.

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