Path Enumeration Research Articles

Path enumeration using flow graphs

This paper deals with a simple method of path enumeration in a reliability logic diagram using the flow graph concept. The reliability graph is reduced using a transformation known as NDT: an example illustrates the method.

Estimation following sequential tests

SUMMARY A method is given for obtaining confidence intervals following sequential tests. It involves defining an order relation among points on the stopping boundary and computing the probability of a deviation more extreme in this order relation than the observed one. Particular attention is given to the case of a normal mean with known or unknown variance. A comparison with the customary fixed sample size interval based on the same data is given. The purpose of this paper is to describe a method for obtaining confidence intervals following sequential tests. An order relation is defined among the points on the stopping boundary of the test. The confidence limits are determined by finding those values of the unknown parameter for which the probabilities of more extreme deviations in the order relation than the one observed have prescribed values. To facilitate understanding the proposed procedures, most of the paper is restricted to estimating the mean of a normal population with known variance following the class of sequential tests recommended by Armitage (1975) for clinical trials. The case of unknown variance is discussed briefly in ? 4. It is easy to see that the proposed method is valid more generally, although the probability calculations required to implement it depend on the specific parent distribution and stopping rule. A closely related method was proposed by Armitage (1958), who studied the case of binomial data numerically by enumeration of sample paths. Let xl, x2, ... be independent and normally distributed with unknown mean ,u and known variance cr2. Let sn =x1 + .. . + xn, and for given b > 0 consider the stopping rule

Regular Algebra Applied to Path-finding Problems

In an earlier paper, one of the authors presented an algebra for formulating and solving extremal path problems. There are striking similarities between that algebra and the algebra of regular languages, which lead one to consider whether the previous results can be generalized—for instance to path enumeration problems—and whether the algebra of regular languages can itself be profitably used for the general study of path-finding problems. This paper gives affirmative answers to both these questions.

A probabilistic multipath traffic assignment model which obviates path enumeration

FOLLOWING A BRIEF DISCUSSION OF SHORTCOMINGS OF ALL-OR- NOTHING ASSIGNMENT AND OF PRIOR EFFORTS IN MULTIPATH ASSIGNMENT, TWO ALGORITHMS ARE PRESENTED, BOTH OF WHICH YIELD RESULTS WITH THE ABOVE PROPERTIES. THE ALGORITHMS DIFFER IN THEIR DEFINITION OF A REASONABLE PATH AND IN THE NUMBER OF TIMES EACH IS EXECUTED TO ASSIGN ALL TRIPS FROM A GIVEN ORIGIN NODE. THE FIRST USES MORE RESTRICTIVE DEFINITION THAN THE SECOND AND REQUIRES A SINGLE EXECUTION FOR EACH DISTINCT ORIGIN/DESTINATION NODE PAIR. THE SECOND, A MINOR VARIATION OF THE FIRST, USES A RELAXED DEFINITION OF A REASONABLE PATH, WHICH ALLOWS THE SIMULTANEOUS ASSIGNMENT OF ALL TRIPS ORIGINATING AT A GIVEN NODE IN A SINGLE EXECUTION. COMPARED WITH THE FIRST ALGORITHM, THE LATTER PARALLEL MULTIPATH ALGORITHM IS MUCH MORE EFFICIENT COMPUTATIONALLY, BUT IS LESS DISCRIMINATING IN ITS SELECTION OF PROBABLE PATHS. (A).

Generalized Reliability Function for Systems of Arbitrary Configurations

In this note we consider a special system that always requires more than one elementary path between its input and output terminals to perform its job. Since the special system may consist of components that are not series-parallel connected, the generalized reliability function for systems of parallel components introduced by Chan is insufficient for studying the reliability of this class of systems. However, an even more generalized reliability function for the special system can be obtained by generalizing the path enumeration approach (path-tracing method). The conventional reliability function and the generalized reliability function of Chan are special cases of the one derived in this note.

Circuit, cutset and path enumeration, and other applications of edge-numbering convention

New methods employing an edge-numbering convention are proposed for listing all circuits and cutsets of an arbitrary graph, which are an improvement over the ones recently given by Maxwell and Reed. Storage of a master path matrix, whereby the all-path matrix between any two nodes of a given graph can be derived by simple modification, is discussed. A technique for generation of all trees, including a given set of edges and some significant properties of master path matrices, are also discussed. The algorithms proposed are all suitable for computer adoption.

On Lattice Paths with Several Diagonal Steps

In this note we consider the enumeration of unrestricted and restricted minimal lattice paths from (0, 0) to (m, n), with the following (μ + 2) moves, μ being a positive integer. Let the line segment between two lattice points on which no other lattice point lies be called a step. A lattice path at any stage can have either (1) a vertical step denoted by S0, or (2) a diagonal step parallel to the line x = ty (t = 1,…, μ), denoted by St, or (3) a horizontal step, denoted by Sμ+1.

Enumeration of paths in digraphs

The solution of the problem of enumeration of then-paths in a digraph has so far been attempted through an indirect approach of enumerating the redundant chains. The approach has yielded an algorithm for determination of the general formula for the matrix of redundantn-chains and also a partial recurrence formula for the same. This paper presents a direct approach to the problem. It gives a recurrence relation expressing the matrix ofn-paths of a digraph in terms of the matrices of (n − 1)-paths of its first-order subgraphs. The result is exploited to give an algorithm for computing the matrix ofn-paths. The algorithm is illustrated with a 6 × 6 matrix.