Tail bound for the minimal spanning tree of a complete graph

Jeong Han Kim; Sungchul Lee

doi:10.1016/S0167-7152(03)00208-6

Tail bound for the minimal spanning tree of a complete graph

Jeong Han Kim, Sungchul Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

Suppose each edge of the complete graph K_n is assigned a random weight chosen independently and uniformly from the unit interval [0,1]. A minimal spanning tree is a spanning tree of K_n with the minimum weight. It is easy to show that such a tree is unique almost surely. This paper concerns the number N_n(α) of vertices of degree α in the minimal spanning tree of K_n. For a positive integer α, Aldous (Random Struct. Algorithms 1 (1990) 383) proved that the expectation of N_n(α) is asymptotically γ(α)n, where γ(α) is a function of α given by explicit integrations. We develop an algorithm to generate the minimal spanning tree and Chernoff-type tail bound for N_n(α).

Original language	English
Pages (from-to)	425-430
Number of pages	6
Journal	Statistics and Probability Letters
Volume	64
Issue number	4
DOIs	https://doi.org/10.1016/S0167-7152(03)00208-6
Publication status	Published - 2003 Oct 1

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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abstract = "Suppose each edge of the complete graph Kn is assigned a random weight chosen independently and uniformly from the unit interval [0,1]. A minimal spanning tree is a spanning tree of Kn with the minimum weight. It is easy to show that such a tree is unique almost surely. This paper concerns the number Nn(α) of vertices of degree α in the minimal spanning tree of Kn. For a positive integer α, Aldous (Random Struct. Algorithms 1 (1990) 383) proved that the expectation of Nn(α) is asymptotically γ(α)n, where γ(α) is a function of α given by explicit integrations. We develop an algorithm to generate the minimal spanning tree and Chernoff-type tail bound for Nn(α).",

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AB - Suppose each edge of the complete graph Kn is assigned a random weight chosen independently and uniformly from the unit interval [0,1]. A minimal spanning tree is a spanning tree of Kn with the minimum weight. It is easy to show that such a tree is unique almost surely. This paper concerns the number Nn(α) of vertices of degree α in the minimal spanning tree of Kn. For a positive integer α, Aldous (Random Struct. Algorithms 1 (1990) 383) proved that the expectation of Nn(α) is asymptotically γ(α)n, where γ(α) is a function of α given by explicit integrations. We develop an algorithm to generate the minimal spanning tree and Chernoff-type tail bound for Nn(α).

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