On the Value of a Random Minimum Weight Steiner Tree

Authors:
Béla Bollobás;David Gamarnik;Oliver Riordan;Benny Sudakov
Affiliations:
University of Memphis, Department of Mathematical Sciences, USA and Institute for Advanced Study, Princeton, NJ, 08540, USA;IBM T. J. Watson Research Center, Department of Mathematical Sciences, USA;Trinity College, Department of Mathematical Sciences, UK;Princeton University, Department of Mathematics, USA and Institute for Advanced Study, Princeton, NJ, 08540, USA
Venue:
Combinatorica
Year:
2004

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Abstract

Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1+o(1))(k-1)(log n-log k)/n when k =o(n) and n→∞.