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Showing papers on "Spanning tree published in 1973"


Journal ArticleDOI
01 Jan 1973-Networks
TL;DR: The problem of allocating cost in a spanning tree network is considered and methods are suggested that are preferred given different emphases among the criteria for such a function.
Abstract: The problem of allocating cost in a spanning tree network is considered. A number of possible schemes are surveyed, and critically analyzed. Methods are suggested that are preferred given different emphases among the criteria for such a function.

162 citations


Journal ArticleDOI
01 Jan 1973-Networks
TL;DR: The capacitated minimum spanning tree is an offspring of the Minimum spanning tree and network flow problems and has application in the design of multipoint linkages in elementary teleprocessing tree networks.
Abstract: The capacitated minimum spanning tree is an offspring of the minimum spanning tree and network flow problems. It has application in the design of multipoint linkages in elementary teleprocessing tree networks. Some theorems are used in conjunction with Little's branch and bound algorithm to obtain optimal solutions. Computational results are provided to show that the problem is tractable.

83 citations


Proceedings ArticleDOI
01 Oct 1973
TL;DR: There is an ordering of the nodes of a flow graph G which topologically sorts the dominance relation and can be found in 0(edges) bit vector steps and it follows that there is a very simple bit propagation algorithm which also uses the above ordering, and is at least as good as the interval algorithm for solving all known global data flow problems.
Abstract: There is an ordering of the nodes of a flow graph G which topologically sorts the dominance relation and can be found in 0(edges) steps. This ordering is the reverse of the order in which a node is last visited while growing any depth-first spanning tree of G. Moreover, if G is reducible, then this ordering topologically sorts the "dag" of G. Thus, for a reducible flow graph (rfg) there is a simple algorithm to compute the dominators of each node in 0(edges) bit vector steps.The main result of this paper relates two parameters of an rfg. If G is reducible, d is the largest number of back edges found in any cycle-free path in G, and k is the length of the interval derived sequence of G, then k≥d. From this result it follows that there is a very simple bit propagation algorithm (indeed, the obvious one) which also uses the above ordering, and is at least as good as the interval algorithm for solving all known global data flow problems such as "available expressions" and "live variables."

68 citations


Proceedings ArticleDOI
David W. Hightower1
25 Jun 1973
TL;DR: This paper represents a fairly extensive survey of the literature on the interconnection problem and presents algorithms forPin Assignment, Layering, Ordering, Wire List Determination, Spanning Trees, Rectilinear Steiner Trees, and Wire Layout.
Abstract: This paper represents a fairly extensive survey of the literature on the interconnection problem. The topics covered are: Pin Assignment, Layering, Ordering, Wire List Determination, Spanning Trees, Rectilinear Steiner Trees and Wire Layout. In addition, several new ideas are presented which could provide for better wire layout. Algorithms are presented in a way that makes them easy to understand and hence, to discuss and apply. Formal statement of the algorithms can be found in the references cited.

59 citations


Proceedings ArticleDOI
01 Jan 1973
TL;DR: It is pointed out that all of these algorithms fall into the class of minimum spanning tree problems, constrained by traffic or response time requirements, and most of the algorithms can be unified into a modified Kruskal'sminimum spanning tree algorithm.
Abstract: The problem of designing minimum cost multidrop lines which connect remote terminals to a concentrator or a central data processing computer is studied. In some cases, optimal solutions can be obtained by using either linear integer programming or a branch-bound method. These approaches are not practical since they lack flexibility and require an enormous amount of computer time for most practical problems. As a consequence, heuristic algorithms have been developed by various authors. In this paper, we point out that all of these algorithms fall into the class of minimum spanning tree problems, constrained by traffic or response time requirements. The difference between them is mainly the sequential order with which a branch or a line is selected into the tree. Without the constraints, all algorithms converge to a minimum spanning tree. With the constraints, they form different sub-trees. Most of the algorithms can be unified into a modified Kruskal's minimum spanning tree algorithm.In the modified algorithm, a weight is associated with each terminal. Let Wi be the weight associated with terminal i, and dij be the cost for the line directed from terminal i to terminal j. When the algorithm fetches the cost for the line, it replaces it with dij - wi In some cases, Wi's need to be readjusted in the middle of the algorithm. The difference between all existing heuristic algorithms is in the way wi's are defined. If wi is zero for all i, the algorithm reduces to the unmodified Kruskal's algorithm; if wi is set to zero whenever a line incident to terminal i is selected as a tree branch, the algorithm reduces to Prim's minimum spanning tree algorithm.An extension of the algorithm to the solution of an associated problem of partitioning the terminals with respect to a predetermined set of concentrators, multiplexers, terminal interface processors, or central computers is also derived.The efficiency of an algorithm depends greatly on how it is implemented. The computational complexity of the unified algorithm is in the order of N2 log N for the most general case, where N is the number of terminals. By using good heuristics, it reduces to K1 K log N + K2 N, where K1 and K2 are constants, for many practical applications. The algorithm has been applied to large networks with over 1,000 terminals, yielding excellent results and using only 15 seconds of computer time on a CDC 6600 computer.Designs obtained by using different wi's are compared.

36 citations


Proceedings ArticleDOI
15 Oct 1973

16 citations


Journal ArticleDOI
TL;DR: In this paper, a recurrence relation and an explicit solution for the number of spanning trees in a multigraph wheel with unequal numbers of spokes and rim edges are given, which can be identified as special cases of the general enumeration.
Abstract: A recurrence relation and an explicit solution for the number of spanning trees in a multigraph wheel with unequal numbers of spokes and rim edges are given. Previous results can be identified as special cases of the general enumeration and the solution for one new specific case is included.

10 citations


Proceedings ArticleDOI
15 Oct 1973

6 citations




Book ChapterDOI
07 May 1973
TL;DR: This work deals with trees, showing how the algorithm for finding the shortest spanning tree of a graph can be modified to handle some problems of this kind efficiently.
Abstract: When two weighting figures per arc must be considered in optimizing a network one has either to combine them in a single performance factor or to optimize the network with respect to one of them while respecting some constraint on the other. This work deals with trees, showing how the algorithm for finding the shortest spanning tree of a graph can be modified to handle some problems of this kind efficiently.

Journal ArticleDOI
TL;DR: Doulliez and Rao as mentioned in this paper presented algorithms that solve two flow problems for a single source, multi-terminal network, where the demand at each sink is a nondecreasing, linear function of t. The proofs are based on the well known fact that a network possesses only a finite number of different spanning trees.
Abstract: In their paper [Doulliez, P. J., M. R. Rao. 1971. Maximal flow in a multi-terminal network with any one are subject to failure. Management Sci.18 1, September 48--58.], Doulliez and Rao present algorithms that solve two flow problems for a single source, multi-terminal network. The first problem that they solve is the construction of a flow that maximizes the value of t, where the demand at each sink is a nondecreasing, linear function of t. Given such a flow, the second problem that they solve is the construction of a flow that maximizes the value of t when the capacity of an arc is reduced. This paper supplies a finiteness proof for the first algorithm and sketches a finiteness proof for the second algorithm. The proofs are based on the well-known fact that a network possesses only a finite number of different spanning trees.

Journal ArticleDOI
TL;DR: The cable-communications systems, m -center, and minimum spanning-tree problems are contained within a general graph optimization-synthesis problem where the total number of vertices is estimated from the length of the spanning tree.
Abstract: The cable-communications systems, m -center, and minimum spanning-tree problems are contained within a general graph optimization-synthesis problem. In the minimum spanningtree problem, a decomposition technique is used in constructing minimal-length trees. This problem is related to the m -center problem where the total number of vertices is estimated from the length of the spanning tree. Both results, vertex estimate and decomposition, are utilized in the cable-communications systems problem. A procedure and example are provided for estimating the dollar cost of a cable-communications system.



Journal ArticleDOI
TL;DR: A generating function that facilitates the counting and listing of the spanning trees in many classes of graphs is presented, with examples of its application.
Abstract: A generating function that facilitates the counting and listing of the spanning trees in many classes of graphs is presented, with examples of its application.

Proceedings ArticleDOI
27 Aug 1973
TL;DR: In this paper, an algorithm for finding the fundamental cycle set from a spanning tree and a branch and bound option which will find the longest cycle in a graph is presented, which is based on a more efficient branching procedure than was used previously.
Abstract: An algorithm is presented which generates, without duplication, all the simple cycles of an undirected graph. The algorithm employs the concept of a fundamental cycle set and is based on a more efficient branching procedure than was used previously.1,2 As a consequence both execution time and storage requirements are greatly reduced. Also included is an improved method for finding the fundamental cycle set from a spanning tree, and a branch and bound option which will find the longest cycle in a graph.