Terminal Capacity Matrix Research Articles

Synthesis of flow networks minimizing the sum of edge capacities

AbstractSeveral synthesis methods to construct a flow network with the minimum sum of edge capacities for a given terminal capacity matrix have already been presented. However, there are networks that cannot be constructed by the known method.For the case where the given terminal capacity matrix is called elementary terminal capacity matrix, a synthesis method is presented that can construct any undirected flow network with the minimum sum of edge capacities. This paper is based on that synthesis method and presents a synthesis method that can construct any undirected flow network with the minimum sum of edge capacities for a given terminal capacity matrix with that matrix as the terminal capacity matrix.

Synthesis of flow networks minimizing the sum of edge capacities

AbstractSeveral methods to construct a flow network minimizing the sum of edge capacities which have the given terminal capacity matrix are already known. However, there exist some networks that cannot be constructed by known methods.This paper considers the case where the given terminal capacity matrix is an elementary terminal capacity matrix. A method is proposed to construct a flow network with the minimum sum of edge capacities, which has the given matrix as the terminal capacity matrix. Using this method of construction, any flow network with the minimum sum of edge capacities that has the given elementary terminal capacity matrix can be constructed.

Synthesis of a flow network with a given centrality on each vertex

AbstractThis paper considers the problem of constructing a flow network when the centrality at each vertex is given. Here, attention is given to a centrality function such that the centrality at a vertex under consideration is the sum of maximum flow values between it and all other vertices. It is a representative centrality function among those representing the centrality of each vertex in an undirected flow network in which edges have capacity. First, we introduce a necessary and sufficient condition wherein a given sequence is a centrality sequence for a flow network. Next, we present a procedure for determining a terminal capacity matrix of a flow network with a given centrality sequence. Furthermore, we determine a terminal capacity matrix of a flow network with a given centrality sequence such that the sum of edge capacity is minimum. From these discussions we can see that the problem of constructing a flow network when a centrality is given, can be reduced to the previously known problem of constructing a flow network when a terminal capacity matrix is given.

Realization of a symmetric terminal capacity matrix by a tree with the minimum diameter

AbstractTrees realizing a terminal capacity matrix (TCM) have a unique layer structure. An algorithm making use of this layer structure is presented for construction of a tree which realizes the TCM with the smallest possible diameter. This minimum‐diameter tree has layers of stars. An upper bound on the minimum diameter is given for a prescribed number of vertices. Also presented are algorithms to construct a star and a tree of diameter 3 which realize the TCM as a minimum requirement matrix.

Probabilistic Communication Net as a Nonoriented Graph

This paper proves a necessary and sufficient realizability condition of the s-expected terminal capacity matrix of a probabilistic communication net. The proof is based on the successive reduction processes of a matrix, which provide a technique for designing a tree topology and determing edge capacities and reliabilities that satisfy the matrix.

On a relationship between terminal capacity and impedance matrices

It is shown that the terminal capacity matrix of a nonoriented communication net and the open-circuit impedance matrix of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port resistor network without internal loops and with a star port configuration have identical properties. A new set of necessary and sufficient conditions characterizing such matrices is presented.

Realizability of communication nets: An application of the Zadeh criterion

The concepts of pseudo-Boolean matrix multiplication and pseudo-Boolean matrix adjoint are reviewed. For a given symmetric matrix T to be realizable as the terminal capacity matrix of an unoriented communication net, it is necessary and sufficient for T to be idempotent with respect to multiplication. Another equivalent condition is for T to be self-adjoint.

Synthesis of nonoriented 2n-terminal communication nets with prescribed terminal capacity matrices of two-port flows

The realization of nonoriented communication nets satisfying requirements of two commodities is investigated in this paper. A necessary condition for a requirement matrix to be a terminal capacity matrix of a communication net is found. A method of synthesizing a 2n -terminal net with the use of a special configuration is presented. The results based on graph theory are more convenient than those of linear programming. Whenever a realization is obtained, it is an optimum realization. The procedures which convert a 2n -terminal net into a 2(n+ 1) -terminal net are also discussed.

Survivable communication networks and the terminal capacity matrix

The problem of constructing networks that are "survivable" with respect to branch damage is considered. A square redundancy matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R = [r_{i,j}]</tex> is specified. Algorithms are given to construct a graph with minimum number of branches so that for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i, j</tex> there are 1) at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r_{i,j}</tex> undirected branch disjoint paths between the ith and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</tex> th vertices, or 2) there are exactly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r_{i,j}</tex> undirected branch disjoint paths between the ith and the jth vertices. These algorithms are closely related to the optimal realization of terminal capacity matrices. Two of the algorithms are extended to the optimal realization of terminal capacity matrices for symmetric or pseudosymmetric graphs.

Necessary and Sufficient Conditions for Realizability of Vertex-Weighted Communication Nets

A counter example is given to show that the conditions for a matrix to be realizable as the terminal-capacity matrix of a vertex-weighted communication net, which were given in 1962, are only necessary but not sufficient. Necessary and sufficient conditions for realizability are proposed and proved. Refinements in the description of a terminal-capacity matrix are presented and a new synthesis algorithm is introduced. Finally, conditions for special realizability classes of graphs are obtained as immediate consequences of the necessary and sufficient conditions.

Algorithms to Realize Directed Communication Nets

Necessary and sufficient conditions are formulated for a real square unsymmetric matrix R to be the terminal capacity matrix of a directed communication net. These conditions lead to a practical set of realizability tests and are used to derive easily tested sufficient conditions for realizability. The sufficient conditions include several previously known realizability conditions as special cases and answer some hitherto unsolved problems concerning the range of applicability of a number of realization schemes.

Author's reply to 'On the Maximum Number of Different Entries in the Terminal Capacity Matrix of Oriented Communication Nets'

Author's reply to 'On the Maximum Number of Different Entries in the Terminal Capacity Matrix of Oriented Communication Nets'

On the Maximum Number of Different Entries in the Terminal Capacity Matrix of Oriented Communication Nets

On the Maximum Number of Different Entries in the Terminal Capacity Matrix of Oriented Communication Nets

Synthesis of Radio-Communication Nets

A radio-communication net is a communication network in which links between stations have unlimited capacity to transmit information. Such a network can be represented by a graph with weights on the vertices which correspond to the capacities of individual stations. The capacity of a radio-communication net of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> stations is described by a matrix of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> called a terminalcapacity matrix. A necessary and sufficient condition for a matrix to be realizable as the terminal-capacity matrix of a radio-communication net is found. A simple algorithm for synthesizing a radio-communication net of a given terminal-capacity matrix, which is proved to have minimum sum of vertex capacities and minimum number of branches, is obtained. A theorem concerning all possible structures of a radio-communication net with the same terminal-capacity matrix is established.

Analysis and Synthesis Techniques of Oriented Communication Nets

An oriented communication net is a communication network in which channel capacities between pairs of terminals are not symmetrical. Such a system can be represented by an oriented graph. The concept of a minimum-valued cut is used to determine the terminal-capacity matrix which gives the maximum possible communication between any ordered pair of terminals. Necessary conditions for a terminal capacity matrix to be realizable as an oriented communication network are obtained. These conditions are shown to be sufficient when the order of the terminal capacity matrix is three or less. They are not sufficient for higher-order cases in general. Synthesis techniques based on matrix partition and matrix addition are used to realize any three-by-three terminal capacity matrix with extensions to some higher-order cases.

Synthesis of a Communication Net

A systematic method is given for the realization of communication nets from their terminal capacity matrices. It is shown that this procedure results in a net whose total branch capacity is minimum for all nets satisfying the same terminal capacity matrix. It is also shown that when the terminal capacity matrix is indeterminate, then, for a given total branch capacity, the total terminal capacity is highest when all terminal capacities are made equal.

Terminal and Branch Capacity Matrices of a Communication Net

A communication net consists of branches representing communication channels with the weight of each being a positive real number which represents the capacity of transferring information through the (called a branch capacity). The terminal capacity between the vertices i and j of a communication net is the capacity of transferring information between the vertices i and j by considering the net as a whole. To indicate the terminal capacities between all possible pairs of vertices in a net, a terminal capacity matrix is defined. Then the necessary and sufficient conditions for a terminal capacity matrix are given. To represent the structure of a communication net, a capacity matrix is defined. Then the synthesis of a communication net from a given terminal capacity matrix is to obtain a capacity matrix from the given terminal capacity matrix.