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Showing papers on "Longest path problem published in 1979"


Journal ArticleDOI
01 Sep 1979-Networks
TL;DR: The study shows that the procedures examined indeed exert a powerful influence on solution efficiency, with the identity of the best dependent upon the topology of the network and the range of the arc distance coefficients.
Abstract: : This paper examines different algorithms for calculating the shortest path from one node to all other nodes in a network. More specifically, we seek to advance the state-of-the-art of computer implementation technology for such algorithms and the problems they solve by examining the effect of innovative computer science list structures and labeling techniques on algorithmic performance. The study shows that the procedures examined indeed exert a powerful influence on solution efficiency, with the identity of the best dependent upon the topology of the network and the range of the arc distance coefficients. The study further discloses that the shortest path algorithm previously documented as the most efficient is dominated for all problem structures by the new methods, which are sometimes an order of magnitude faster. (Author)

237 citations


Journal ArticleDOI
A. Nadas1
01 May 1979
TL;DR: The probability distribution of the critical pathlength turns out to be a solution of an unconstrained minimization problem, which can be recast as a convex programming problem with linear constraints.
Abstract: A solution is offered to the problem of determining a probability distribution for the length of the longest path from source (start) to sink (finish) in an arbitrary PERT network (directed acyclic graph), as well as determining associated probabilities that the various paths are critical ("bottleneck probabilities"). It is assumed that the durations of delays encountered at a node are random variables having known but arbitrary probability distributions with finite expected values. The solution offered is, in a certain sense, a worst-case bound over all possible joint distributions of delays for given marginal distributions for delays. This research was motivated by the engineering problem of the timing analysis of computer hardware logic block graphs where randomness in circuit delay is associated with manufucturing variations. The probability distribution of the critical pathlength turns out to be a solution of an unconstrained minimization problem, which can be recast as a convex programming problem with linear constraints. The probability that a given path is critical turns out to be the Lagrange multiplier associated with the constraint determined by the path. The discrete version of the problem can be solved numerically by means of various parametric linear programming formulations, in particular by one which is effciently solved by Fulkerson's network flow algorithm for project cost curves.

53 citations


Journal ArticleDOI
TL;DR: This work defines 2-bicritical graphs and gives several characterizations of them, and shows that almost all graphs are 2- bicritical and hence the linear relaxation almost never helps for large random graphs.
Abstract: The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical and hence the linear relaxation almost never helps for large random graphs.

42 citations



Journal ArticleDOI
TL;DR: A unified algebraic theory for a class of path problems such as that of finding the shortest or, more generally, the k shortest paths in a network; the enumeration of elemementary or simple paths inA graph is developed.

20 citations


Journal ArticleDOI
TL;DR: This work considers the class of 2-trees and presents a linear time algorithm for finding minimum dominating cycles of such graphs and stresses the use of a particular representation of these graphs called a recursive representation, and some linear operations on directed trees associated with these graphs.
Abstract: We consider the class of 2-trees and present a linear time algorithm for finding minimum dominating cycles of such graphs. We stress the use of a particular representation of these graphs called a recursive representation, and some linear operations on directed trees associated with these graphs.

19 citations


Book ChapterDOI
TL;DR: This chapter presents a survey of several more "classic” results for some of the more “standard” problem formulations, and an estimate of the worst-case running time of the algorithm is presented in the chapter.
Abstract: Publisher Summary A truly remarkable variety of discrete optimization problems can be formulated and solved as shortest path or network flow problems or can be solved by procedures that employ shortest path or network flow algorithms as subroutines. It follows that these network computations are among the most fundamental and important in the entire area of discrete optimization. This chapter presents a survey of several more “classic” results for some of the more “standard” problem formulations. In each case, an estimate of the worst-case running time of the algorithm is presented in the chapter. Bellman's equations is also discussed; virtually, all the methods for finding the length of a shortest path between two specified nodes embed the problem in the larger problem of finding the lengths of shortest paths from an origin to each of the other nodes in the network.

18 citations


01 Jan 1979
TL;DR: In this article, it was shown that the problem of finding minimum cutsets in general directed graphs is known to be NP-complete, and that it can be found in linear time.
Abstract: The analysis ofmany processes modeled by directed graphs requires the selection of a subset of vertices which cut all the cycles in the graph. Reducing the size of such a cutset usually leads to a simpler and more efficient analysis, but the problem of finding minimum cutsets in general directed graphs is known to be NP-complete. In this paperwe show that in reducible graphs (and thus in almost all the "practical" flowcharts of programs), minimum cutsets can be found in linear time. We further show that the linear algorithm can check its own applicability to a given graph, thus eliminating the need of prechecking (in nonlinear time) whether it is reducible or not. An immediate application of this result is in program verification systems based on Floyd's inductive assertions method. 1. Motivation. A directed graph is often used as a path-generating device, which models the succession of events (in the form of edge traversals) that can take place in some process. Two common examples are the graph representations of finite state machines (with edges labeled by symbols from some alphabet) and of flowcharts of computer programs (with edges labeled by instructions). Finite directed graphs which do not contain cycles can describe only finitely many paths, each of which contains finitely many edges, and thus the path-analysis of these graphs is usually straightforward. The analysis becomes qualitatively different in the presence of cycles, since the number and length of the paths need not be finite any longer. However, in many cases the path-analysis of arbitrary graphs can be reduced to that of cycle-free graphs by selecting an appropriate subset of vertices (called cutpoints) such that any cycle in the graph contains at least one cutpoint. These cutpoints dissect the graph in a natural way into cycle-free components, which can be analyzed separately. All that remains to be done is to relate the overall behavior of the original graph to that of its components, and this is usually done by some kind of induction. An important concrete example of such an analysis is Floyd's method for proving the partial correctness of computer programs (Floyd (1967)). Since execution sequences of instructions may be arbitrarily long (or infinite), one uses the selected cutpoints in the flowchart in order to "chop" them into subsequences ofbounded size. If the correctness of the specifications attached to the cutpoints is preserved along any such subsequence, one can infer the overall correctness of the program by induction of the number of subsequences. Graphs may have many sets of cutpoints, all of which are useful in principle (one example is the set of all the vertices pointed to by backward edges in a depth-first search; the potential redundancy in this choice of cutpoints is demonstrated in Fig. 1). In many cases, the number of cutpoints selected has a strong influence on the complexity of the subsequent analysis. For example, if each cutpoint gives rise to an equation (where the equated quantities may be numbers, logical formulas, or sets of_strings), and the time required in order to solve n such simultaneous equations is a rapidly growing function of n, then minimizing the number n of cutpoints can be very desirable.

16 citations


Journal ArticleDOI
TL;DR: An optimal one-line diagram generator algorithm is presented and the problem is posed and solved as a longest path problem.
Abstract: An optimal one-line diagram generator algorithm is presented. The corresponding cctputer program drives the output device to draw a one-line diagram which is optimal in the sense that for a given drawing area, the minimum distance between elements of the diagram is a maximum. The computer program needs little additional information about the topology of the network to that normally supplied to a load flow program. The problem is posed and solved as a longest path problem. Examples of results of this program coupled to a load flow program are presented.

8 citations


01 Jul 1979
TL;DR: In this paper, a general predictor-corrector method is described for following the curve, which is a globalization of the classical Davidenko approach and is shown to follow the path to any desired degree of accuracy and is convergent.
Abstract: : The problem of finding one or all solutions to systems of equations, equilibrium, fixed points, or to dynamical systems is considered. In the last few years, a new method has emerged for solving this problem. The idea is to start at a given solution of a simpler problem and to follow a path of solutions as the path parameter (and hence, the problem) is gradually changed. This path is proved to exist via topological approaches and is shown to lead to the 'right' place. In this paper, a general predictor-corrector method is described for following the curve. It is a globalization of the classical Davidenko approach. It is shown that the method can follow the path to any desired degree of accuracy and is convergent.

6 citations



Journal ArticleDOI
TL;DR: In this paper, a simple method of path enumeration in a reliability logic diagram using the flow graph concept is presented, where the reliability graph is reduced using a transformation known as NDT.

Journal ArticleDOI
TL;DR: In this paper, the authors pointed out that the use of nodes can result in a network path being incorrectly identified as a path of maximum length, which can lead to the path of the maximum length being incorrectly selected.
Abstract: The critical path in a PERT/CPM network is determined in various textbooks by finding the path with associated minimal slack for the network arcs or by finding the path with associated minimal slack for the network nodes. This paper points out that the use of nodes can result in a network path being incorrectly identified as a path of maximum length.