Path Vectors Research Articles

Maximal Level Minimal Path Vectors of a Two-Terminal Undirected Network

A two-terminal flow network is usually defined as a directed graph, or a digraph; but there are a number of applications where it is more natural to use an undirected graph. This paper analyzes the connection between minimal path vectors and flow functions in undirected networks, which supports the development of an efficient algorithm that solves the problem of finding the set of all such vectors.

Evaluation of Algorithms for Identification of Minimal Cut Vectors and Minimal Path Vectors in Multi-State Systems

Minimal Cut Vectors (MCVs) and Minimal Path Vectors (MPVs) are one of the key concepts of reliability analysis. They allow us to estimate system availability or to analyze influence of individual system components on the entire system. However, the main problem of their use, especially in reliability analysis of complex systems, lies in their identification. Several algorithms have been proposed to solve this task. Some of the most universal ones are based on logical differential calculus. These algorithms use integrated direct partial logic derivatives to find situations that can correspond to the MCVs (MPVs) and a special type of logic conjunction to select only those situations that really agree with the MCVs (MPVs). In this paper, we summarize the ideas behind these algorithms in more formal way and present results of some experiments performed to study their time complexity.

Ordering Heuristics for Reliability Evaluation of Multistate Networks

This paper develops ordering heuristics to improve the efficiency of reliability evaluation for multistate two-terminal networks given all minimal path vectors ( $ d $ -MPs for short). In the existing methods, all $d$ -MPs are treated equally. However, we find that the importance of each $d$ -MP is different, and different orderings affect the efficiency of reliability evaluation. Based on the above observations, we introduce the length definitions for $d$ -MPs in a multistate two-terminal network, and develop four ordering heuristics, called O1, O2, O3, and O4, to improve the efficiency of the Recursive Sum of Disjoint Products (RSDP) method for evaluating network reliability. The results show that the proposed ordering heuristics can significantly improve the reliability evaluation efficiency, and O1 performs the best among the four methods. In addition, an ordering heuristic is developed for the reliability evaluation of multistate two-terminal networks given all minimal cut vectors ( $d$ -MCs).

Search for all d-MPs for all d levels in multistate two-terminal networks

A general method for reliability evaluation of multistate network is using minimal path vectors. A minimal path (MP) vector to system state d is called a d-MP. Most reported works on generating d-MPs are for a particular d value. However, if all d-MPs for all possible integer d values are required, we need to call such methods multiple times with respect to all d values. A more efficient method is desirable to generate all d-MPs. In this paper, we develop a recursive algorithm based on breadth-first search to search for all the d-MPs for all possible d values. The relationships among d-MPs for different d levels are revealed. Each d-MP candidate can be generated by a combination of one (d-1)-MP and the vector form of one binary minimal path. Thus, we can use binary MPs as building blocks to generate 2-MP candidates, and use 2-MPs and binary MPs as building blocks to generate 3-MP candidates … and so forth. When the d-MPs with respect to the maximum d value have been found, all the d-MPs for all possible d values are obtained. A heuristic for pre-processing the MPs is proposed to improve the efficiency of the algorithm. Through computational experiments, it is found that the proposed algorithm is more efficient than existing algorithms for finding all d-MPs for all possible d values. In addition, we show that the proposed algorithm can also be used to generate a subset of d-MPs for all or some d values given a subset of MPs. The generated subset of d-MPs can be used for lower reliability bound evaluation.

Optimal representation of piecewise Hölder smooth bivariate functions by the Easy Path Wavelet Transform

The Easy Path Wavelet Transform (EPWT) (Plonka, 2009) [26] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we aim to provide a theoretical understanding of the performance of the EPWT. In particular, we derive conditions for the path vectors of the EPWT that need to be met in order to achieve optimal N-term approximations for piecewise Hölder smooth functions with singularities along curves.

Deriving the Upper Bound of the Number of Sensors Required to Know All Link Flows in a Traffic Network

It is demonstrated that the minimum number of sensors required to know all link flows in a traffic network can be determined only if path information is available. However, not all paths need to be enumerated but, at most, a small subset defining the rank rw of the link-path incidence matrix W. If this rank for a reduced subset of paths is already m - n, where m and n are the number of links and noncentroid nodes, respectively, we can conclude that m - n sensors are sufficient. It is also shown that the formulas providing the dependent link flows in terms of the independent link flows can be obtained by the node-based or path-based approaches with the same results only when rw = m - n. Finally, an algorithm to obtain the small subsets of linearly independent path vectors is given. The methods are shown by a parallel network example and the Ciudad Real and Cuenca networks, for which the savings in link counts with respect to the m - n bound are larger than 16%. The corresponding savings in path enumeration are larger than 80%.

Wavelet Shrinkage on Paths for Denoising of Scattered Data

We propose a new algorithm for denoising of multivariate function values given at scattered points in $${\mathbb{R}^{d}}$$ . The method is based on the one-dimensional wavelet transform that is applied along suitably chosen path vectors at each transform level. The idea can be seen as a generalization of the relaxed easy path wavelet transform by Plonka (Multiscale Model Simul 7:1474–1496, 2009) to the case of multivariate scattered data. The choice of the path vectors is crucial for the success of the algorithm. We propose two adaptive path constructions that take the distribution of the scattered points as well as the corresponding function values into account. Further, we present some theoretical results on the wavelet transform along path vectors in order to indicate that the wavelet shrinkage along path vectors can really remove noise. The numerical results show the efficiency of the proposed denoising method.

Secure routing in MANETs using local times

We present a new approach to secure routing in mobile ad-hoc networks based solely on the relative transmission times of overhead packets. Unlike most previous works aimed at securing route computation, we eliminate a key vulnerability (explicitly stated routing metrics) altogether. We introduce the Secure Time-Ordered routing Protocol (STOP), which uses time-based orderings to ensure the establishment of multiple loop-free paths between a source and a destination. STOP is the first routing protocol to use performance-based path selection without source routing, path vectors, or complete topology information, making it far more efficient that similar approaches. We prove that adversaries cannot take any action to manipulate the time-based ordering so as to unfairly gain control of the forwarding topology and, by design, nodes which drop data packets will be avoided. Furthermore, at convergence, traffic load is evenly distributed over the well-performing paths, so adversaries cannot gain complete control over the data flow through temporary good behavior. Simulation results show that the countermeasures in STOP are effective against a variety of attacks from independent and colluding adversaries, and that this improved security does not come at the expense of routing performance.

Bounds for the Availabilities of Multistate Monotone Systems Based on Decomposition into Stochastically Independent Modules

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

Bounds for the Availabilities of Multistate Monotone Systems Based on Decomposition into Stochastically Independent Modules

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

Reliability evaluation of network flows with stochastic capacity and cost constraint

This paper addresses flow networks whose arcs have finite and independent random capacities. The reliability of such a system, that is the probability that d units of flow can be transmitted from the source node to the sink node with total cost at most equal to c , can be computed in terms of Minimal Path Vectors, named as ( d , c )-MPs. In this paper, first, a method for computing the upper and lower bounds of the capacity cost of the system is proposed. The presented method shows how to compute a ( d , c )-MP with the minimum capacity cost. Subsequently, we develop an algorithm, based on the computed bounds, to generate all ( d , c )-MPs and then, the system reliability is calculated in terms of such ( d , c )-MPs. The computational complexity of the proposed algorithm is also presented. At the end the solution procedure is illustrated by an example.

Test Vector Generation for Post-Silicon Delay Testing Using SAT-Based Decision Problems

Debugging and speed-binning a fabricated design requires a pattern-dependent timing model to generate patterns, which static timing analysis is incapable of providing. To address these issues, we propose a timing analysis tool that integrates a pattern-dependent delay model into its analysis. Our approach solves for the delay by using the concept of circuit unrolling and formulation of timing questions as decision problems for input into a satisfiability (SAT) solver. We generate a critical path and input vectors that stimulate it, taking into account pattern-dependent effects such as data-dependent gate delays and multiple-inputs switching. The effectiveness and validity of the proposed methodology is illustrated through experiments on various benchmark circuits and comparisons directly with SPICE.

An Efficient Multistate Multivalued Decision Diagram-Based Approach for Multistate System Sensitivity Analysis

Multistate systems (MSS) are systems in which the system and its components are characterized by multiple states or performance levels. Component importance or sensitivity analysis facilitates the identification of vulnerabilities within the system, and aids in the quantification of criticalities of the system components. Multistate component importance analysis poses unique challenges to existing methods that are primarily based on binary-state applications. This paper presents an analytical method based on multistate multivalued decision diagrams (MMDD) for multistate component importance analysis. The contribution of this work is two-fold: 1) a novel, efficient algorithm for directly generating an MMDD model from multistate capacity network specifications without inefficient enumeration of multistate minimal path or cut vectors; and 2) an efficient, exact MMDD-based approach for evaluating MSS reliability and importance measures. The advantages of the proposed method are illustrated through a comparison with existing methods, and through detailed analyses of three case studies.

A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform

The easy path wavelet transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image approximation that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure.

Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.

Extended dominance and a stochastic shortest path problem

In the context of stochastic networks, we study the problem of finding a path P that combines in a reasonable way the mean m ( P ) and variance v ( P ) of its length. Specifically we study a separable objective function that combines these two path measures: namely, z ( P ) = f ( m ( P ) ) + g ( v ( P ) ) , where f is an increasing convex function and g is an increasing concave function. A new type of dominance ( e-dominance), stronger than the standard form of dominance, is then introduced, and it is shown to satisfy a certain form of Bellman's optimality principle. This means that it is possible to modify existing label-setting and label-correcting methods by using e-dominance, and without sacrificing optimality. Computational experience with these enhanced labeling algorithms has been promising. Test results for a variety of sample problems show that the e-dominance criterion can often significantly reduce the number of nondominated path vectors, compared to the standard dominance criterion. We observe a consequent reduction in both computation time and storage requirements.

A Classification Tree Based Approach for the Development of Minimal Cut and Path Vectors of a Capacitated Network

This paper presents a holistic method that links together Monte-Carlo simulation, exact algorithms, and a data mining technique, to develop approximate bounds on the reliability of capacitated two-terminal networks. The method uses simulation to generate network configurations that are then evaluated with exact algorithms to investigate if they correspond to a network success or failure. Subsequently, the method implements commercially available software to generate a classification tree that can then be analyzed, and transformed into capacitated minimal cut or path vectors. These vectors correspond to the capacitated version of binary minimal cuts & paths of a network. This is the first time that these vectors are obtained from a decision tree, and in this respect, research efforts have been focused on two main directions: 1) deriving an efficient yet intuitive approach to simulate network configurations to obtain the most accurate information; and given that the classification tree method could for some applications provide imperfect or incomplete information without the explicit knowledge of the reliability engineer, 2) understand the relationship between the obtained vectors, and the real capacitated minimal cut & path vectors of the network, and its reliability. The proposed method has been tested on a set of case networks to assess its validity & accuracy. The results obtained show that the technique described is effective, simple, and widely applicable.

An efficient method for reliability evaluation of multistate networks given all minimal path vectors

The multistate networks under consideration consist of a source node, a sink node, and some independent failure-prone components in between the nodes. The components can work at different levels of capacity. For such a network, we are interested in evaluating the probability that the flow from the source node to the sink node is equal to or greater than a demanded flow of d units. A general method for reliability evaluation of such multistate networks is using minimal path (cut) vectors. A minimal path vector to system state d is called a d-MP. Approaches for generating all d-MPs have been reported. Given that all d-MPs have been found, the issue becomes how to evaluate the probability of the union of the events that the component state vector is greater than or equal to at least one of the d-MPs. There is a need for a more efficient method of determining the probability of this union of events. In this paper, we report an efficient recursive algorithm for this union probability evaluation based on the Sum of Disjoint Products (SDP) principle, and name it the Recursive Sum of Disjoint Products (RSDP) algorithm. The basic idea is that, based on the SDP principle and a specially defined “maximum” operator, “⊕”, the probability of a union with L vectors can be calculated via calculating the probabilities of several unions with L−1 vectors or less. The correctness of RSDP is illustrated. The efficiency of this algorithm is investigated by comparing it with an existing algorithm that is generally accepted to be efficient. It is found that RSDP is more efficient than the existing algorithm when the number of components of a system is not too small. RSDP provides us with an efficient, systematic and simple approach for evaluating multistate network reliability given all d-MPs.

A Heuristic Approach for Constrained Redundancy Optimization in Multi-state Systems

This paper proposes an efficient heuristic approach to solving the constrained redundancy optimization problem in multi-state systems (MSS) with multi-state components using minimal path vectors. A discrete multi-state model is considered, where the system state depends on the discipline of the elements' interaction in the system. When the multi-state nature of the system is considered, exact solution methodologies e.g. Dynamic, Integer Programming are no longer valid. The proposed heuristic offers an efficient and straightforward analysis. To illustrate the simplicity and ease of the application of the algorithm, solutions of a flow network problem with linear constraint and, bridge structure with linear as well as nonlinear constraints are obtained. The results would be applicable to multi-state design problems in real life.

A generalized multistate-based path vector approach to multistate two-terminal reliability

The two-terminal reliability problem assumes that a network and its elements are either in a working or a failed state. However, many practical networks are built of elements that may operate in more than two states i.e., elements may be degraded but still functional. Multistate two-terminal reliability at demand level d (M2TR d ) can be defined as the probability that the system capacity generated by multistate components is greater than or equal to a demand of d units. This paper presents a fully multistate-based algorithm that obtains the multistate equivalent of binary path sets, namely, Multistate Minimal Path Vectors (MMPVs), for the M2TR d problem. The algorithm mimics natural organisms in the sense that a select number of arcs inherit information from other specific arcs contained in a special set called the “primary set.” The algorithm is tested and compared with published results in the literature. Two features of the algorithm make it relevant: (i) unlike other approaches, it does not depend on an a priori knowledge of the binary path sets to obtain the MMPVs; and (ii) the use of an information sharing approach and network reduction technique significantly reduce the number of vector analyses needed to obtain all the component levels that guarantee system success. Additionally, the complexities associated with the computation of reliability are discussed. A Monte Carlo simulation approach is used to obtain an accurate estimate of actual M2TR values based on MMPVs. Examples are used to validate the algorithm and the simulation procedure.