On the shortest spanning subtree of a graph and the traveling salesman problem

Abstract

Relationships observed between variables are inextricably linked to the ways in which the variables are measured. In an earlier paper it was argued that the development of a rigorous and consistent system of measurement is a necessary, although not a sufficient, condition for the discovery of mathematical relationships in human geography [ 5 ] . This note will examine the measurements presently in use in the subject to substantiate the claim that the types of variables we investigate do not facilitate mathematical synthesis. Consideration of geographical variables leads to questioning the extent to which the discovery of mathematical relationships is an appropriate goal.The primitive terms from which all measurements can be derived are known as dimensions. It was suggested in [ 5 ] that a working set of primary dimensions for human geography might be mass M , distance L, time T, value $, population N, and information 1. From these primitives it is possible to construct most, if not all, of the variables of interest. Population density, for example, is population divided by the product of two distances (NL-2), land rent is value per area per time ($L-'T-l), and so on. Analyzing variables in this manner emphasizes the compatibility of their measurements and clarifies how it is that many different quantities may combine together in an elegant and parsimonious mathematical relationship. According to the principle of dimensional homogeneity, all quantities that are added, subtracted, or equated must have the same dimensions or, in other words, must be measured in the same units [ 6 ] . Population, for example, cannot equal area, but can equal area multiplied by population density (L2.NL-'= N). Population density here is a constant that has the effect of linking the measurement of population and area and so makes possible a mathematical relation between them. A collection of variables that are made up of different combinations of the same primary dimensions is rich in theoretical possibilities. On the other hand, a field in which the most commonly used measures are not compatible with each other is unlikely to develop mathematical theory. R. M. Haynes is lecturer in environmental sciences, Unioersity of East Anglia. 0 1978 Ohio State University Press 0016-7363/78/0778-0288$00.50/0 GEOGRAPHICAL ANALYSIS, vol. X, no. 3 (July 1978)

Chow–Liu trees are sufficient predictive models for reproducing key features of functional networks of periictal EEG time-series

“…2.6 w.r.t. E becomes a minimum spanning tree problem, for which efficient solution methods exists, most notably Kruskal's algorithm (Kruskal, 1956) and Prim's algorithm (Prim, 1957 Pearson correlation coefficient matrices were derived from the raw, continuous iEEG data, discretized iEEG data and the distributions P ij (x i , x j ), (i, j) ∈ V 2 inferred from the CL-trees. Throughout this paper we will denote by sCC = (scc ij ) the ordinary (signed) Pearson correlation-coefficient matrix (but with zero main diagonal) and with CC = (|scc ij |) the corresponding absolute value Pearson correlation-coefficient matrix, from which functional networks were constructed.…”

Large Quasi-Tree Drawing: A Neighborhood Based Approach

“…The filtration in that algorithm is made by first computing a minimum spanning tree of the graph using the well-known Kruskal's algorithm [23] and then rooting that tree on the graph center. This rooted tree allows to know the order in which vertices of the graph must be inserted in the layout.…”

Data Mining with Graphical Models

“…However, for the simple relational example, i.e., the relation R ABC (with or without the tuple (a 4 , b 3 , c 2 )) we may simply choose the well-known Kruskal algorithm [Kruskal 1956], which determines an optimum weight spanning tree. As edge weights we may choose the relative number of possible value combinations (and determine a minimum weight spanning tree) or the Hartley information gain (and determine a maximum weight spanning tree).…”

“…given edge weights. Nevertheless, the well-known Kruskal algorithm [Kruskal 1956] is guaranteed to construct an optimum weight spanning tree and it is clearly efficient. (There is also an even more efficient algorithm for this task [Prim 1957].…”

“…All possible (undirected) edges over the set U of attributes used to describe the domain under consideration are evaluated with an evaluation measure ([Chow and Liu 1968] used mutual information 24 ). Then an optimum weight spanning tree is constructed with either the (well-known) Kruskal algorithm [Kruskal 1956] or the (somewhat less well-known) Prim algorithm [Prim 1957] (or any other greedy algorithm for this task).…”