Steele, J.M., Shepp, L.A. J.M. Eddy, W. (1987): On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab., 24, 809--826. Home/Search Document Not in Database Summary Related Articles Check |
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On Properties of Geometric Random Problems in the Plane - Jaillet (1994) (Correct)
....algorithms such as Kruskal s. This scheme defines an asymptotically optimal algorithm with excellent theoretical convergence properties coupled with very fast practical convergence (with 100 points, the probability that the heuristic does not give the optimal solution is below 0. 001) Also, in [36], the authors prove that for any independent and uniform random variables fX i : 1 i 1g in [0; 1] 2 , the number of vertices of degree k in the MST through fX 1 ; Xng is asymptotic to a constant ff k times n with probability one. In the case k = 1 (i.e. for the number of leaves of ....
J. Steele, L. Shepp, and W. Eddy. 1986. On the Number of Leaves of a Euclidean Minimal Spanning Tree. Journal of Applied Probability, 24, 809--826.
Cube versus Torus Models and the Euclidean Minimum.. - Patrick Jaillet August (1992) (1 citation) (Correct)
....distributed over [0; 1] d . Then there are positive constants ff k;d such that lim n 1 n Gamma1 V (t) k (X (n) lim n 1 n Gamma1 V (c) k (X (n) ff k;d (a.s. 4. 17) Proof: The existence of the constants verifying the second equality was proved in Steele et al. [10]. Also, if F(X (n) denotes the set of edges of K (t) mst (X (n) that crosses the boundary of the d cube (see the proof of Theorem 1) it is easy to see that, with probability one, we have K (t) mst (X (n) n F(X (n) ae K (c) mst (X (n) 4.18) Now consider any ....
....by d (log n=n) 1=d . In fact, it is not difficult to show (see for example [4] that, for a Poisson point process n with intensity n times the Lebesgue measure on [0; 1] d , the growth of the largest edge is Theta( log n=n) 1=d ) almost surely. Also, in Section 4, we have noted that in [10], the authors prove that for any independent and uniform random variables fX i : 1 i 1g in [0; 1] d , d 2, the number of vertices of degree k in the MST through fX 1 ; X n g is asymptotic to a constant ff k;d times n with probability one. In the case k = 1 and d = 2 (i.e. for the ....
Steele, J., L. Shepp, and W. Eddy. 1986. On the Number of Leaves of a Euclidean Minimal Spanning Tree. Journal of Applied Probability, 24, 809--826.
The Objective Method: Probabilistic Combinatorial.. - Aldous, Steele (2002) (2 citations) Self-citation (Steele) (Correct)
....example. Nevertheless, there are surely cases where subadditive methods have been used in the past, but where the objective now provides an easier or more informative approach. Confirmed cases on this list include the limit theory for MST with power weighted edges [58] and MST vertex degrees [61]; and, with some systematic e#ort, the list can probably be extended to include the theory of Euclidean semi matchings [59] optimal cost triangulations [56] and the K median problem [35] Moreover, there are many cases where the objective method quickly gives one the essential limit theory, yet ....
Steele, J.M., Shepp, L.A. J.M. Eddy, W. (1987): On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab., 24, 809--826.
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