# Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem

• Anand Srivastav
• Sören Werth
Conference paper

DOI: 10.1007/978-3-540-77050-3_41

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4855)
Cite this paper as:
Srivastav A., Werth S. (2007) Probabilistic Analysis of the Degree Bounded Minimum Spanning Tree Problem. In: Arvind V., Prasad S. (eds) FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2007. Lecture Notes in Computer Science, vol 4855. Springer, Berlin, Heidelberg

## Abstract

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 find 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 affirmative 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 satisfies
$$\lim_{n\rightarrow \infty} \frac{L_{bMST}(X_1,\dots,X_n)}{n^{(d-1)/d}}=\alpha(L_{bMST},d)\int_{[0,1]^d} f(x)^{(d-1)/d} dx \text{ c.c.,}$$
where α(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 α(LbMST,d) = α(LTSP,d). We also show concentration of LbMST around its mean and around Open image in new window. The result of this paper may spur further investigation of probabilistic spanning tree problems with degree constraints.