Transitive distance is an ultrametric with elegant properties for clustering. Conventional transitive distance can be found by referring to the minimum spanning tree (MST). We show that such distance metric can be generalized onto a minimum spanning random forest (MSRF) with element-wise max pooling over the set of transitive distance matrices from an MSRF. Our proposed approach is both intuiti...

In their seminal paper on geometric minimum spanning trees, Monma and Suri [11] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O(2n2) × O(2n2) and the authors conjectured that an improvement below cn×cn is not possible, for some constant c > 0. In this paper, we show how to construct MST...

In this work we address the Quadratic Minimum Spanning Tree Problem (QMSTP) known to be NP-hard. Given a complete graph, the QMSTP consists of determining a minimum spanning tree (MST) considering interaction costs between pairs of edges to be modelled. A Lagrangian relaxation procedure is devised and an efficient local search algorithm with tabu thresholding is developed. Computational experim...

Consider a connected r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0,1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then mst(G) ∼ r ζ(3), where ζ(3) = ∑∞ j=1 j −3 ∼ 1.202. Se...

In this survey paper, we discuss the development of the Generalized Minimum Spanning Tree Problem, denoted by GMSTP, and we focus on the integer programming formulations of the problem. The GMSTP is a variant of the classical minimum spanning tree problem (MST), in which the nodes of an undirected graph are partitioned into node sets (clusters) and we are looking for a minimum cost tree spannin...

We introduce the weak gap property for directed graphs whose vertex set S is a metric space of size n. We prove that, if the doubling dimension of S is a constant, any directed graph satisfying the weak gap property has O(n) edges and total weight O(log n) · wt(MST (S)), where wt(MST (S)) denotes the weight of a minimum spanning tree of S. We show that 2-optimal TSP tours and greedy spanners sa...

Most practical techniques for locating remote objects in a distributed system su er from problems of scalability and locality of reference. We have devised the Arrow distributed directory protocol, a scalable and local mechanism for ensuring mutually exclusive access to mobile objects. This directory has communication complexity optimal within a factor of (1+MST-stretch(G))=2, where MST-stretch...

For a set P ⊆ R2 with 2 ≤ n = |P | < ∞ we prove that mst(P ) bb(P ) ≤ 1 √2 √ n + 32 where mst(P ) is the minimum total length of a rectilinear spanning tree for P and bb(P ) is half the perimeter of the bounding box of P . Since the constant 1 √ 2 in the above bound is best-possible, this result settles a problem that was mentioned by Brenner and Vygen (Networks 38 (2001), 126-139).

In many network applications the computation takes place on the minimum-cost spanning tree (MST ) of the network G; unfortunately, a single link or node failure disconnects the tree. The All Node Replacements (ANR) problem is the problem of precomputing, for each node u in G, the new MST should u fail. The simpler problem dealing with single edge failures is called All Edge Replacements (AER). ...

Transport in weighted networks is dominated by the minimum spanning tree (MST), the tree connecting all nodes with the minimum total weight. We find that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality--the number of times a node (or link) is used by transport paths. One component, superhighways, is the in...