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Tail bound for the minimal spanning tree of a complete graph

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  • Kim, Jeong Han
  • Lee, Sungchul

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([alpha]) of vertices of degree [alpha] in the minimal spanning tree of Kn. For a positive integer [alpha], Aldous (Random Struct. Algorithms 1 (1990) 383) proved that the expectation of Nn([alpha]) is asymptotically [gamma]([alpha])n, where [gamma]([alpha]) is a function of [alpha] given by explicit integrations. We develop an algorithm to generate the minimal spanning tree and Chernoff-type tail bound for Nn([alpha]).

Suggested Citation

  • Kim, Jeong Han & Lee, Sungchul, 2003. "Tail bound for the minimal spanning tree of a complete graph," Statistics & Probability Letters, Elsevier, vol. 64(4), pages 425-430, October.
    • Handle: RePEc:eee:stapro:v:64:y:2003:i:4:p:425-430
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