We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline g...

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...

This paper proposes a time-eecient distributed solution for the matroid basis completion problem. The solution is based on a technique called elimination upcast, enabling us to reduce the amount of work necessary for the upcast by relying on the special properties of matroids. As an application, it is shown that the algorithm can be used for correcting a minimum weight spanning tree computed fo...

We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the strict Hadwiger number of the unit ball, i.e., the largest number of unit vectors such that the distance between any two is larger than 1.

Considering the model of computing under uncertainty where element weights are uncertain but can be obtained at a cost by query operations, we study the problem of identifying a cheapest (minimumweight) set among a given collection of feasible sets using a minimum number of queries of element weights. For the general case we present an algorithm that makes at most d · OPT + d queries, where d i...

The minimal weight of a spanning tree in a complete graph K n with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph.

The goals of the course are threefold. First, the student should take from the course a general understanding of the major events, persons, and themes of United States history from contact to the Civil War. The course is designed both to serve as an overview of two and a half centuries of American history and to allow the in-depth exploration of some of the most important issues of that time pe...

We present runt pruning, a new clustering method that attempts to find modes of a density by analyzing the minimal spanning tree of a sample. The method exploits the connection between the minimal spanning tree and nearest neighbor density estimation. It does not rely on assumptions about the specific form of the data density (e.g., normal mixture) or about the geometric shapes of the clusters,...