Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem

In the b-degree constrained Euclidean minimum spanning tree problem (bMST) we are given n points in [0;1]d and a degree constraint b  2. The aim is to nd a minimum weight spanning tree in which each vertex has degree at most b. In this paper we analyze the probabilistic version of the problem and prove in armative the conjecture of Yukich stated in 1998 on the asymptotics of the problem for uniformly (and also some non-uniformly) distributed points in [0;1]d : the optimal length LbMST (X1;:::;Xn) of a b-degree constrained minimal spanning tree on X1;:::;Xn given by iid random variables with values in [0;1]d satises lim n3I LbMST (X1;:::;Xn) n(d 1)=d = (LbMST ;d) Z [0;1]d f(x)(d 1)=d dx c.c., where (LbMST ;d) is a positive constant, f is the density of the absolutely continuous part of the law of X1 and c.c. stands for complete convergence. In the case b = 2, the b-degree constrained MST has the same asymptotic behavior as the TSP, and we have (LbMST ;d) = (LT SP ;d). We also ...

• Type: Conference
• Year: 2007
• Where: FSTTCS
• Authors: Anand Srivastav, Sören Werth
• Keywords: Euclidean Minimum Spanning Tree, FSTTCS 2007, Spanning Tree, Spanning Tree Problems