We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

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A note on two problems in connexion with graphs

Minimum spanning trees (MST) and single linkage cluster analysis (SLCA) are explained and it is shown that all the information required for the SLCA of a set of points is contained in their MST. Known algorithms for finding the MST are discussed. They are efficient even when there are very many points; this makes a SLCA practicable when other methods of cluster analysis are not. The relevant computing procedures are published in the Algorithm section of the same issue of Applied Statistics. The use of the MST in the interpretation of vector diagrams arising in multivariate analysis is illustrated by an example.

Minimum Spanning Trees and Single Linkage Cluster Analysis

The theoretical basis of sequential circuit synthesis is developed, with particular reference to the work of D. A. Huffman and E. F. Moore. A new method of synthesis is developed which emphasizes formal procedures rather than the more familiar intuitive ones. Familiarity is assumed with the use of switching algebra in the synthesis of combinational circuits.

A method for synthesizing sequential circuits

We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100034095

The shortest path through many points

It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the problem. They have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century. We shall explore and compare these works and their motivations, and relate them to the most recent advances on the minimum spanning tree problem.

On the History of the Minimum Spanning Tree Problem

In the construction of a digital computer in which high-frequency circuitry is used, it is desirable and often necessary when making connections between terminals to minimize the total wire length in order to reduce the capacitance and delay-line effects of long wire leads. Presented here are two methods for systematically selecting the shortest connections from a list of possible connections to obtain a minimum total wire length. The special problem considered here is the following: Given a number of terminals, fixed in space, which must be electrically connected together, what procedure will provide the minimum wire length? A proper pattern of connections is one in which there exists one and only one path, either direct or through other terminals, from each terminal to every other terminal and in which there are no loops created by redundant connections. Figure 1 shows two possible proper patterns for connecting four terminals. Problems of this type have been considered in topological areas of mathematics , more particularly in the theory of graphs [1]. In accordance with the prevailing terminology, the following definitions will be used hereafter in this paper. A terminal either connected or mlconnected will be referred to as a node. The direct connection between two nodes is a branch, the magnitude of which is the distance between the nodes. A path between two nodes is a connection consisting of one or more branches. A graph is a structure of nodes connected pairwise by one or more branches. A tree is a graph having one and only one path between every two nodes. It has previously been referred to as a proper pattern. A minimum proper pattern, i.e., a proper pattern where the sum of the wire lengths is a minimum, will be called a minimum tree. A subtree is a tree comprising k of n nodes where k < n. To connect electrically n nodes into a tree, exactly (n-1) branches are necessary [1]. If more than (n-1) branches are used there will be redundant connections and loops will be formed. If less than (n-1) branches are used, not all of the nodes will be interconnected. If exactly (n-1) branches are used, but incorrectly , both loops and unconnected nodes result. To produce a minimum tree, it is conceivably possible to investigate all of the possible trees that exist for n nodes. It can be shown [2, 3] that the number of …

Formal Procedures for Connecting Terminals with a Minimum Total Wire Length

The PILOT data processor is a high-speed multiple computer system, more than 100 times faster than SEAC. It contains three interconnected computers for rapid processing of data, and also contains multiple input-output channels for rapid transfer of data into and out of the system. All of these units operate concurrently in a coordinated fashion. A summary description is given of the over-all logical plan of the system, including the principal characteristics of the first computer, the second computer, the third computer, the internal controls, and the external controls.The system combines internal processing capabilities that are fast and versatile with external communication capabilities that are exceptionally flexible. This combination permits the internal power of the machine to be exploited readily by the outside world,—either by other automatic devices or by human operators.

PILOT—A New Multiple Computer System

In the course of developing the system plans for the DYSEAC and the SEAC, certain standard methods and procedures were evolved for producing a large-scale digital computer design These standard procedures cover, first, the development of system specifications, second, the development of functional plans, and finally the development of wiring plans The later stages of these procedures are reducible to sequences of simple steps, capable of being systematically formulated in explicit terms The similarity between these procedures and many of the data-processing procedures commonly being executed by present-day computers suggests that, with further development of these design techniques, the wiring plans for new computer systems might well be produced by existing digital machines

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System design of the SEAC and DYSEAC

When n terminals are electrically connected so that they are at the same potential, the resulting pa t te rn of connections constitutes a graph. A graph is defined [1] as a number of nodes connected pairwise by branches. The language of the theory of graphs is employed in this paper for the sake of compatibil i ty with existing literature. Thus, a node corresponds to a terminal and a branch between two nodes corresponds to a direct connection between two terminals. Priifer has shown [2] tha t n nodes can be connected in a tree in exactly the same number of ways as (n 2) elements can be independently chosen with replacement from among n elements, namely in n (~-:) ways. (A tree is defined as a graph where one and only one pa th exists between any two of its nodes [1].) To effect his proof he developed a method for expressing a particular tree of these n nodes by means of a unique symbol containing (n 2) elements, where the elements represent nodes. Following the presentation of Prtifer 's method, the authors present two additional methods for designating trees.

A. Weinberger

Papers

Formal Procedures for Connecting Terminals with a Minimum Total Wire Length

PILOT—A New Multiple Computer System

System design of the SEAC and DYSEAC

Symbolic Designations for Electrical Connections