Showing papers in "Networks in 1971"

The steiner problem in graphs

Abstract: An algorithm for solving the Steiner problem on a finite undirected graph is presented. This algorithm computes the set of graph arcs of minimum total length needed to connect a specified set of k graph nodes. If the entire graph contains n nodes, the algorithm requires time proportional to n3/2 + n2 (2k-1 - k - 1) + n(3k-1 - 2k + 3)/2. The time requirement above includes the term n3/2, which can be eliminated if the set of shortest paths connecting each pair of nodes in the graph is available.

Steiner's problem in graphs and its implications

Abstract: A graph theoretic version of Steiner's problem in plane geometry is described. An approach for solving this problem, related to Melzak's solution to Steiner's problem, is presented. The problems of finding “shortest route” and “minimal spanning tree” in graphs become special cases of the Steiner's problem in graphs. It is shown that a solution to this problem also provides us with a solution to the problems of finding a minimum externally stable set and a maximum internally stable set in a graph.

On some techniques useful for solution of transportation network problems

Abstract: This paper presents an efficient algorithm for solving transportation problems. The improvement over the existing algorithms of the “primal-dual” type [3], [5] consists in modifying the “potential-raising” stages of the solution process in such a way that negative-cost arcs are removed so that the Dijkstra's algorithm may be applied. Especially, the algorithm requires at most n3 additions and comparisons when applied to an n-by-n assignment problem, as compared with the theoretical upper bound proportional to n4 for the number of such operations required by currently available methods. Furthermore, auxiliary techniques of simplifying the original network by means of “reduction” and “induction” are also introduced as useful tools to treat large-scale problems and specially-structured problems with.

Minimum cost routing for dynamic network models

Abstract: This paper develops techniques which can be applied to long range planning studies for the domestic long haul communications network. The problem studied is how to select a path through the network for each point-to-point demand for communications channels, so that the total network cost is minimized. The problem is to minimize total network cost, subject to multicommodity flow requirements and concave link cost functions. Finding an exact solution is difficult because of the concavity of the cost functions and the complexity (100 to 200 nodes and 200 to 300 links) of the network structure. A specialized technique is developed to provide locally optimal solutions to the problem, one of minimizing a concave function over a convex constraint set. When the link cost displays a fixed charge, a modification of the iterative algorithm provides acceptable a modification of the iterative algorithm provides acceptable solutions. Even though the global optimum cannot be found amidst the immense number of local optima, several sample problems demonstrate the value of the techniques developed. A companion paper will extend this work to the dynamic routing problem, of deciding how to route future demands and install transmission facilities so as to minimize the present worth of expenditures during the study interval. The methods and goals of these modeling efforts will be discussed in the companion paper.

Network reliability analysis: Part I

Abstract: This paper considers networks with randomly failing links and nodes. In Part 1, nodes are assumed to be perfectly reliable. A combinatorial analysis is given when all links have equal reliabilities. Two general simulation methods are described. The first is particularly useful if a wide range of failure probabilities is to be considered. The second combines a combinatorial analysis with stratified sampling to yield major computational savings. Later parts will describe generalizations, decomposition methods for large networks and applications to computer network analysis.

On non‐blocking switching networks

Abstract: A switching network may be informally described as a collection of single-pole, single-throw switches arranged so as to connect a set of terminals called inputs to another set of terminals called outputs. It is non-blocking if, given any set of connections from some of the inputs to some of the outputs, and an idle input terminal x and idle output terminal y, then it is possible to connect x to y without disturbing any of the existing connections. Denote by σ(a, b) the minimal number of switches necessary to connect a inputs to b outputs using a non-blocking network. We are interested in studying the growth of σ(a, a) as a ∞. Results of C. Clos show that σ(a, a) ⩽ C ae2√log a·log 2. We show that σ(a, a) ⩽ 8a(log2a)2.

A simple derivation of edmonds' algorithm for optimum branchings

Abstract: Edmonds [1] has given an algorithm for constructing a maximum-weight branching in a weighted directed graph. His proof that the algorithm is correct is based on linear programming theory, and establishes as a by-product that a certain polyhedron has integer vertices. Here we give a direct combinatorial proof of the correctness of the algorithm.

Routing in computer networks

Abstract: The problem of routing flow in a network of computers is extremely complex This is especially formidable when routing is to be incorporated in iterative analysis and design Among the properties of desirable flow patterns is low average delay from message inception to arrival In this paper, we discuss procedures for minimizing average delay subject to a set of flow constraints Heuristic routing procedures are presented and compared to optimum routing procedures Computational experience is given

Optimal design of centralized computer networks

Abstract: Major design problems for centralized computer networks are link layout and capacity assignment. The objective is to select link locations and capacities so that the average time delay required to transmit a message from any node to the central node does not exceed a specified maximum. The design problem is to find the least cost system which satisfies the time delay constraints for given levels of traffic. In this paper, we describe an algorithm to select globally optimum link capacities for specified tree structures. We also give a heuristic method for finding low cost tree structures. The methods have been programmed and are capable of handling networks with thousands of nodes. In addition, they can consider any finite set of link capacities with an arbitrary cost structure and do not depend on the mathematical model used to calculate average time delay.

Imbalances in k‐colorations

Abstract: 1. INTRODUCTION The following problem is due to Paul Erdijs tll. Color the edges of a complete graph K on n vertices red and blue. What n is the largest t such that we may always find a complete subgraph in which I# red edges-# blue edges1 Lt? We need a more precise and more general formulation. For any set V, define Vk = {W :, w&V, Iwl = k). (1) Note that V2 is the complete graph generated by V. Vk is called the complete k-graph generated by V. The elements of Vk are i~ll~dn~-~~~~~enW~yC~l~~pthe k-edges. A coloring of a set A, 'k : Ak-+ (+l,-1). The values +l,-1 may be thought of as Red and Blue. The sub-script k indicates a function on k-edges and will be dropped when there is no confusion. The function g k induces another function, also denoted by gk, on the subsets of A given by Set gk@) =

Optimal ranking of tournaments

Abstract: Let us be given a tournament Tn (i-e., a complete directed graph) on n players. It is natural to look for a ranking (linear order) L on the n players that best reflects the tournament result. Thinking of L as a complete directed transitive graph we may use IE(T) fl E(L) I, where E(T) = edge set of T, as a measure of how good a fit L is. We ask, what is the worst case. Set f (n) = min max IE(T ) nE(L) 1 . n Tn

A generalized upper bounding algorithm for multicommodity network flow problems

Abstract: : An algorithm for solving min cost or max flow multicommodity flow problems is described. It is a specialization of the simplex method, which takes advantage of the special structure of the multicommodity problem. The only non- graph or non-additive operations in a cycle involve the inverse of a working basis, whose dimension is the number of currently saturated arcs. Efficient relations for updating this inverse are derived.

The processor-sharing queueing model for time-shared systems with bulk arrivals

Abstract: We consider a model which is applicable to time-multi-plexed systems, such as multiplexed communication channels and time-shared computing facilities. In this (processor-sharing) queueing model, all jobs currently in the system share equally the processing capability of the server. In this paper, we investigate the processor-sharing model for the case of bulk arrivals. The mean response time of the system as a function of required service time is derived. An example is given to show the effect of bulk arrivals versus single arrivals for a constant utilization.

Minimal k‐arc connected graphs

Abstract: : A linear graph is k-arc-connected if it is necessary to remove at least k arcs in order to disconnect the graph. This paper solves the problem of determining the fewest number of arcs required in a k-arc-connected graph on n nodes by describing constructions that produce such graphs having kn/2 arcs (for kn even) or kn + l/2 arcs (for kn odd). These results have application to the practical problem of synthesizing minimum cost, 'k-reliable' communication networks.

Two‐way traffic in loop service systems

Abstract: A model for a communication system which consists of a computer and a number of buffered terminals connected by means of a loop channel is analyzed. Data flows in two directions: from the computer to the terminals and from the terminals to the computer. The channel is alternately available to the computer and the terminals to effect these data transfers. The transient behavior of the system is determined and the stationary or limiting expected queue lengths at all terminals are calculated.

An algorithm for certain multi-commodity flow problems

Abstract: In this paper an algorithm is presented for solving a certain class of multi-commodity flow problems. The class of problems considered consist of integer capacity multi-source and multi-sink problems for which each node is a source of sink for all but at most one commodity. The algorithm is described in the case that any edge connecting a source and sink of a commodity can be used for flow of that commodity.

Optimal flow in networks with gains and costs

Abstract: The problem of routing a desired flow at minimum cost through a network in which associated with each arc is a capacity, a linear cost, a fixed cost and a flow multiplier is considered. The equivalence of the problems on a general graph and a bipartite graph is shown; feasibility is discussed and properties of solutions are investigated.

A decomposition approach to nonlinear programs as applied to reservoir systems

Abstract: The problem considered in this paper is the determination of the operating policy over time of a network of water reservoirs or dams arranged according to an arbitrary topology. If streamflow was a known quantity, the problem could be formulated as a large nonlinear program (several hundred nonlinear) constraints as a minimum) and solved by the two methods proposed in this paper. Since streamflow is random, the problem becomes a large stochastic program which can be solved by a procedure given in the literature. The key step in solving this stochastic program is the solution of the nonlinear program by the methods we propose.

A study of the analysis and control of the flow of air traffic: Part III

Abstract: This paper contains the results of an exploratory study aimed at developing a model which could form the basis of a comprehensive analysis of the Air Traffic Control system. This model could be used to evaluate the effects of proposed changes in the system. The results will be presented in three parts. Part I is intended to provide an overview of the results as well as a detailed description of a traffic flow model which computes the means and variances of delays experienced by individual aircraft. Since this model requires as an input a statistical description of the arrival and departure service processes, Part II describes a model which relates these processes to more basic quantities (e.g., aircraft mix, runway configuration). Since the models which are developed represent a physical system, the question of validation arises and is also discussed in Part II. Part III of the paper considers the question of designing a (real-time) flow control system. It is formulated as a problem of minimizing a measure of system delay subject to constraints on the allowable departure and arrival times. Illustrative results are included through-out the paper.