Path Enumeration Research Articles

The path-status graph with application to delay fault simulation

We present an efficient path-delay fault (PDF) simulator that does not involve the enumeration of paths. Our method calculates the exact fault coverage, and identifies all tested faults in any circuit with a large number of paths. We present a new data structure, called the path-status graph (PSG), to efficiently hold the status of each PDF in the circuit, i.e., whether or not the PDF is tested. The keg to this efficiency is in breaking the information into pieces and distributing it over the data structure, and in retaining all or part of the reconverging fan-out structure of the circuit in the PSG. Thus, an exponential number of PDF's can share the same piece of information. Using 1000 random tests, we simulated all of the approximately 10/sup 20/ PDF's in the circuit c6288, and determined that 4.4 billion faults were detected. This number is larger by over three orders of magnitude compared to what was possible with previously reported methods.

A Link-Based Alternative to Bell’s Logit Assignment Algorithm

A Bell’s logit assignment method was recently proposed as an alternative to Dial’s algorithm. While retaining the absence of a need for path enumeration, Bell’s method dispenses with either a forward or a backward pass. It therefore does not require minimum node-to-node cast information beforehand. The aniy constraint on the set of feasibie paths imposed is that there are no loops and/or cycles. However, it is difficult to avoid loaps in real networks. In order to minimize the cycle problems, a link-based alternative to Bell’s logit assignment method is proposed with making use of the volume successive average algorithm. The absence of any efficiency constraint on the set offeasible paths makes the algorithm attractive for use in stochastic user equilibrium assignment. Extensions of the proposed method to limit the number and type of cycles permitted in the feasible path set are also discussed. Numerical examples are given to illustrate the applications of the proposed method and to compare the results wi...

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, \widehat{\rm des}, is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of des on genus zero permutations and Baxter permutations. Our q-enumerative results relate the des statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.

Bicriterion traffic assignment: Efficient algorithms plus examples

This paper describes the mathematical properties and solution algorithm of a stochastic bicriterion equilibrium traffic assignment model in which both criteria may be flow-dependent. Formulated as a variational inequality, the model can reflect that different trip makers—due to habit, taste, or information—react differently to various factors such as travel time, out-of-pocket cost, and traffic congestion. Combining a novel tree builder/trip loader that obviates path enumeration together with Newton's method and linear programming, its reduced simplicial decomposition solution algorithm is fast enough to make the model a practical planning tool.

A stochastic user equilibrium path flow estimator

The paper sets out a path flow estimator suitable for use in conjunction with urban traffic monitoring, control and guidance. Travel time for each link in the network is partitioned into undelayed travel time and delay. The links are assumed to be of two types. For the first type of link, an external estimate of flow and travel time over the estimation interval is provided. The second type of link is characterised by a finite capacity, and delay is incurred where demand would otherwise be in excess of capacity. Demand is determined by a logit route choice model. An equivalent convex programming problem is formulated and an iterative solution procedure is set out. The estimation of the dispersion parameter in the logit model is discussed, and a column generation method to avoid path enumeration is proposed. Diagnostic procedures and a number of other practical enhancements to the procedure, in particular the incorporation of prior information on the relative magnitudes of origin-destination movements, are considered.

A probit-based stochastic user equilibrium assignment model

Stochastic methods of traffic assignment have received much less attention in the literature than those based on deterministic user equilibrium (UE). The two best known methods for stochastic assignment are those of Burrell and Dial, both of which have certain weaknesses which have limited their usefulness. Burrell's is a Monte Carlo method, whilst Dial's logit method takes no account of the correlation, or overlap,between alternative routes. This paper describes, firstly, a probit stochastic method (SAM) which does not suffer from these weaknesses and which does not require path enumeration. While SAM has a different route-finding methodology to Burrell, it is shown that assigned flows are similar. The paper then goes on to show how, by incorporating capacity restraint (in the form of link-based cost-flow functions) into this stochastic loading method, a new stochastic user equilibrium (SUE) model can be developed. The SUE problem can be expressed as a mathematical programming problem, and its solution found by an iterative search procedure similar to that of the Frank-Wolfe algorithm commonly used to solve the UE problem. The method is made practicable because quantities calculated during the stochastic loading process make the SUE objective function easy to compute. As a consequence, at each iteration, the optimal step length along the search direction can be estimated using a simple interpolation method. The algorithm is demonstrated by applying it successfully to a number of test problems, in which the algorithm shows good behaviour. It is shown that, as the values of parameters describing the variability and degree of capacity restraint are varied, the SUE solution moves smoothly between the UE and pure stochastic solutions.

Stochastic User Equilibrium Assignment with Explicit Path Enumeration: Comparison of Models and Algorithms

In this paper a preliminary analysis of alternative models for “feasible” path generation and choice is presented. In particular a k-shortest path multi-criteria model for path enumeration is explored and different choice models (Logit, recently proposed C-Logit and Probit) are tested by comparing SUE assignment link flows with counts on an urban road network. Flows are also compared for more traditional DUE and SUE Probit implicit path enumeration models. The results obtained show that a limited number (4-7) of paths generated with rather “simple” criteria give satisfactory results, SUE with explicit path enumeration is largely comparable with, and in some cases superior to, traditional implicit SUE and DUE models. Explicit path enumeration allow also the specification of more sophisticated non additive attributes in the utility function of route choice models. From the computational point of view the explicit path C-Logit and Probit SUE algorithms are from three to twenty times superior to the implicit Probit SUE assignment.

A DSS for Tactical Planning of Intercity Railway Services

This paper presents a large Decision Support System for the operational planning of rail services named SASM, recently developed by the Italian Railways company “Ferrovie dello Stato”. The first part of the paper describes shortly the basic functionalities of the DSS. The second part of the paper describes the demand model based on a tree-logit service/run/class choice model that explicitly takes into account the desired departure/arrival time; the supply model based on a diachronic (spatial-temporal) network representation of scheduled services; the assignment model which is a Stochastic Network Loading (SNL) model based on explicit (diachronic) path enumeration. At the end some applications of the model system to the Torino-Venezia rail corridor are reported.

Integrated Models for Simulating the Italian Passenger Transport System

Abstract In this paper, a system of models simulating intercity passenger travel demand in Italy is presented. The models are part of a wider project for a Decision Support System at the national level (SIMPT) covering both passenger and freight transport systems.The core of the SIMPT is composed by integrated demand, supply and assignment models. Behavioural demand models, calibrated on a large RP survey, tour frequency, destination, mode and path choices for different purposes, days and periods of the year. Supply models consist in infrastructural (roads, railways) and a service (railway services, extraprovincial buses, airlines) networks. The road assignment model is a pseudo-within-day-dynamic SUE with explicit path enumeration and takes into account the calibrated path choice models for different time slices, while assignment to transit services is a SNL line-based modal.

Performance analysis of the bilayered ShuffleNet and the SR_Net under deflection routing

The bilayered ShuffleNet is a new logical configuration for multihop networks. This configuration uses the same logical configuration as the conventional ShuffleNet in terms of the number of users, number of columns and connectivity between columns: a doubling of connectivity is obtained by allowing each station in a column to be connected to a ‘mirror’ image of stations from the previous column. We compare the performance of the bilayered ShuffleNet with the SR_Net, a modified version of the conventional ShuffleNet, which can have the same connectivity and the same number of users as the bilayered ShuffleNet. These networks will be studied under deflection routing by applying the method of signal flow to path enumeration, and by a Markov chain model of the system.

Performance analysis of embedded software using implicit path enumeration

Embedded computer systems are characterized by the presence of a processor running application-specific dedicated software. A large number of these systems must satisfy real-time constraints. This paper examines the problem of determining the extreme (best and worst) case bounds on the running time of a given program on a given processor. This has several applications in the design of embedded systems with real-time constraints. An important aspect of this problem is determining which paths in the program are exercised in the extreme cases. The state-of-the-art solution here relies on an explicit enumeration of program paths. This solution runs out of steam rather quickly since the number of feasible program paths is typically exponential in the size of the program. We present a solution for this problem that does not require an explicit enumeration of program paths, i.e., the paths are considered implicitly. This solution is implemented in the program cinderella (in recognition of her hard real-time constraint-she had to be back home at the stroke of midnight), which currently targets a popular embedded processor-the Intel i960. The preliminary results of using this tool are also presented here.

Cyclic flows, Markov process and stochastic traffic assignment

Dial's stochastic assignment algorithm restricts the assignment path set to “efficient path.” As a result, it sometimes produces the unrealistic flow pattern that no flow is loaded on some paths where many vehicles are running in reality. To remove the drawback of Dial's algorithm, this paper presents the LOGIT type assignment that does not restrict the assignment paths. We first show the theoretical relation between the proposed model and Sasaki's assignment model through Markov process. This analysis makes it clear that the proposed assignment model can be calculated by some matrix operations. Next, we propose an efficient algorithm that does not require the matrix operation nor path enumeration over a network. The algorithm solves an equivalent program based on the entropy decomposition derived from the Markov property of LOGIT model. Finally, it is shown that the proposed approach can be easily extended to the flow dependent case (i.e. stochastic equilibrium assignment).

Symmetric Sheffer sequences and their applications to lattice path counting

A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks.

Efficient logic-level timing analysis using constraint-guided critical path search

As the operating speed of digital circuits dramatically increases with the advance of VLSI technology, it is becoming more critical to ensure that the circuits are free from timing-related design errors. In a traditional static timing approach nonfunctional paths cannot be distinguished from functional ones since the functionality of a circuit is ignored. This often results in overestimation of circuit delay and can degrade the circuit performance. In today's design methodology where the use of automated logic synthesis and module-based design are popular, circuits with a very large number of nonfunctional (false) paths are common. This paper describes an efficient logic-level timing analysis approach that can provide an accurate delay estimate of a digital circuit which may have many long false paths. By using logic incompatibilities in a circuit as constraints for critical path search, the algorithm determines the longest sensitizable path without explicit path enumeration. Since the number of false paths that can be implicitly eliminated is potentially exponential to the number of path constraints, performance improvement is significant.

Performance analysis of embedded software using implicit path enumeration

Embedded computer systems are characterized by the presence of a processor running application specific dedicated software. A large number of these systems must satisfy real-time constraints. This paper examines the problem of determining the extreme (best and worst) case bounds on the running time of a given program on a given processor. This has several applications in the design of embedded systems with real-time constraints. An important aspect of this problem is determining which paths in the program are exercised in the extreme cases. The state of the art solution here relies on an explicit enumeration of program paths. This runs out of steam rather quickly since the number of feasible program paths is typically exponential in the size of the program. We present a solution for this problem that does not require an explicit enumeration of program paths, i.e., the paths are considered implicitly . This solution is implemented in the program cinderella 1 which currently targets a popular embedded processor --- the Intel i960. The preliminary results of using this tool are also presented here.

A Combined Algorithm for Solving and Calibrating the Stochastic Traffic Assignment Model

This paper studies Fisk's stochastic user equilibrium (SUE) assignment model in which travellers are assumed not to have perfect knowledge of the traffic system. It is shown that the stochastic properties of this model completely depend on the calibration parameter. After giving the searching method of an 'efficient paths' set, a modified combined algorithm for solving and calibrating Fisk's stochastic traffic assignment model is developed. In this algorithm, the path enumeration is achieved automatically in the preliminary phase and renewed in the iterative processes, and the number of alternative paths generated can be controlled by a predetermined function. Two numerical examples are presented to illustrate the model's properties and the algorithm's efficiencies.

Alternatives to Dial's logit assignment algorithm

Two logit assignment methods for transportation networks are proposed as alternatives to Dial's algorithm. While retaining the absence of a need for the enumeration of paths, they dispense with both forward and backward passes. They, therefore, do not require minimum node-to-node cost information beforehand. Both methods admit loops and paths that are otherwise inefficient in the Dial sense, which can arise in practice as a result of driver searching behaviour. The first method considers a finite number of paths and the second method an infinite number of paths in the presence of loops. The absence of any efficiency constraint on the set of feasible paths makes the algorithms attractive for use in stochastic user equilibrium methods or in the approximation of a user equilibrium assignment through stochastic user equilibrium methods. The similarity of the structure of one of the proposed algorithms with that of the Floyd-Warshall shortest path algorithm would allow the two to be combined.

Generation of design verification tests from behavioral VHDL programs using path enumeration and constraint programming

A method for generation of design verification tests from behavior-level VHDL programs is presented. The method generates stimuli to execute desired control-flow paths in the given VHDL program. This method is based on path enumeration, constraint generation and constraint solving techniques that have been traditionally used for software testing. Behavioral VHDL programs contain multiple communicating processes, signal assignment statements, and wait statements which are not found in traditional software programming languages. Our model of constraint generation is specifically developed for VHDL programs with such constructs. Control-flow paths for which design verification tests are desired are specified through certain annotations attached to the control statements in the VHDL programs. These annotations are used to enumerate the desired paths. Each enumerated path is translated into a set of mathematical constraints corresponding to the statements in the path. Methods for generating constraint variables corresponding to various types of carriers in VHDL and for mapping various VHDL statements into mathematical relationships among these constraint variables are developed. These methods treat spatial and temporal incarnations of VHDL carriers as unique constraint variables thereby preserving the semantics of the behavioral VHDL programs. Constraints are generated in the constraint programming language CLP(R) and are solved using the CLP(R) system. A solution to the set of constraints so generated yields a design verification test sequence which can be applied for executing the corresponding control path when the design is simulated. If no solution exists, then it implies that the corresponding path can never be executed. Experimental studies pertaining to the quality of path coverage and fault coverage of the verification tests are presented. >

Path-integral treatment of multi-mode vibronic coupling. II. Correlation expansion of class averages

A path-integral approach to real-time quantum dynamics is presented which is suitable to treat the dynamics of vibronic coupling or spin boson models. In these models the vibrational dynamics is nonseparable as a consequence of the electronic inter-state coupling. The sum over all possible paths in electronic-state space generated by the usual Trotter procedure is expressed in terms of single-mode averages over classes of paths and statistical mode correlations. The averages for classes of a given length can be calculated iteratively from averages over shorter paths. This expansion is formally exact and finite for a finite number of modes. Usually only a limited number of terms has to be evaluated in order to obtain converged results. The scaling of the computational effort with respect to the number of time steps and the number of modes is given by a low-order power law, depending on the chosen class structure and the order of the expansion. The usual time-dependent wave-packet propagation and the full path enumeration, which exhibit an exponential scaling behavior with respect to either the number of modes or the number of time steps, can be considered as opposite limiting cases of the correlation expansion (CE) of the path integral. The convergence of the CE is tested by application to a two-state four-mode model representing S1-S2 vibronic coupling in pyrazine, for which exact references (time-dependent correlation functions) are available. The potential of the CE approximation for the treatment of multi-mode problems is demonstrated by application to an extended 24-mode vibronic-coupling model. This model is suitable to provide a microscopic description of ultrafast optical dephasing processes in large molecules.

Stochastic user equilibrium assignment in networks with queues

A stochastic user equilibrium assignment algorithm is presented for steady state store-and-forward networks. The links of the network have constant travel times and the links or nodes have finite capacities. When capacity is reached, delay sufficient to match demand to the available capacity is generated. It has been shown by others that the equilibrium assignment in networks of this kind is the solution to a particular linear programming problem. By adding an entropy term to the objective function, a convex nonlinear programming problem is formed which yields a stochastic user equilibrium assignment. For the case of link constraints, it is proven that the Lagrange multipliers of both the linear and the non-linear programming problems give the equilibrium delays in the network. The requirements for uniqueness are investigated. Iterative algorithms are formulated for solving the nonlinear programming problem with either link or node constraints and convergence is proven. For networks where path enumeration is likely to be a problem, a column generation technique is proposed. An illustrative example is presented.