Formal Procedures for Connecting Terminals with a Minimum Total Wire Length
TL;DR: Two methods for systematically selecting the shortest connections from a list of possible connections to obtain a minimum total wire length are presented, which will be called a minimum tree.
Abstract: 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 …
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: 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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TL;DR: 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.
Abstract: 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.
1,210 citations
01 Oct 1959
TL;DR: In this paper, it was shown 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 this result was extended to bounded Lebesgue sets in k-dimensional Euclidean space.
Abstract: 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.
902 citations
TL;DR: There are several apparently independent sources and algorithmic solutions of the minimum spanning tree problem and their motivations, and they have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century.
Abstract: 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.
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01 Feb 1956
TL;DR: Kurosh and Levitzki as discussed by the authors, on the radical of a general ring and three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings.
Abstract: 7 A Kurosh, Ringtheoretische Probleme die mit dem Burnsideschen Problem uber periodische Gruppen in Zussammenhang stehen, Bull Acad Sei URSS, Ser Math vol 5 (1941) pp 233-240 8 J Levitzki, On the radical of a general ring, Bull Amer Math Soc vol 49 (1943) pp 462^66 9 -, On three problems concerning nil rings, Bull Amer Math Soc vol 49 (1943) pp 913-919 10 -, On the structure of algebraic algebras and related rings, Trans Amer Math Soc vol 74 (1953) pp 384-409
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