In their seminal paper on Euclidean minimum spanning trees [Discrete & Computational Geometry, 1992], Monma and Suri proved that any tree of maximum degree 5 admits a planar embedding as a Euclidean minimum spanning tree. Their algorithm constructs embeddings with exponential area; however, the authors conjectured that c × c area is sometimes required to embed an n-vertex tree of maximum degree...

We present a general framework for obtaining efficient algorithms for computing minimum spanning trees. We use this framework to derive the classical algorithms of Prim, Kruskal and Bor̊uvka. We then describe the randomized linear-time algorithm of Karger, Klein and Tarjan. The algorithm of Karger, Klein and Tarjan uses deterministic linear-time implementations of a verification algorithm of Kom...

In this paper, the minimum spanning tree problem is investigated on the graph with fuzzy edge weights. The notion of fuzzy α-minimum spanning tree is presented based on the credibility measure, and then the solutions of the fuzzy α-minimum spanning tree problem are discussed under different assumptions. First, we respectively assume that all the edge weights are triangular fuzzy numbers and tra...

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is ultrametric. We show that every ultrametric minimum cost spanning tree game is a cost allocation game on a rooted tree. It follows that there exist optimal algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost s...

We study the problem of approximating MST(P ), the minimum weight spanning tree of a set P of n points in [0, 1], by a spanning tree of some subset Q ⊂ P . We show that if the weight of MST(P ) is to be approximated, then in general Q must be large. If the shape of MST(P ) is to be approximated, then this is always possible with a small Q. More specifically, for any 0 < ε < 1 we prove: (i) Ther...

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametri...

The minimum degree spanning tree problem has been studied extensively. In this paper, we present a polynomial time algorithm for the minimum degree spanning tree problem on directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove the algorithm must reduce a vertex of the maximum degree for each ...

In the b-degree constrained Euclidean minimum spanning tree problem (bMST) we are given n points in [0; 1] 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 a rmative the conjecture of Yukich stated in 1998 on the asymptotics of the problem for uni...

A method is herewith proposed which after analyzing the data matrix in principal components, searches the subspace representing the initial configuration of the inter-point distances by eliminating the “background noise” present in the data considered. In that subspace a factorial minimum spanning tree is built which becomes a reference structure for the design of a synthetic index of the pheno...

Let G = (V, E, w) be an undirected graph with nonnegative edge weight. For any spanning tree T of G, the weight of T is the total weight of its tree edges and the routing cost of T is ∑ u,v∈V dT (u, v), where dT (u, v) is the distance between u and v on T . In this paper, we present an algorithm providing a trade off among tree weight, routing cost and time complexity. For any real number α > 1...