346 research outputs found
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Tours of graphs, digraphs and sequential machines
A tour of a graph (digraph, or sequential machine) is a sequence of nodes from the graph such that each node appears at least once and two nodes are adjacent in the sequence only if they are adjacent in the graph. Finding the shortest tour. of a graph is known to be an NP-complete problem. Several theorems are given that show that there are classes of graphs in which the shortest tour can be found easily. For more general graphs, we present approximating algorithms for finding short tours. For undirected graphs, the approximating algorithms give tours at worst a constant times the length of the shortest tour. For directed graphs, the size of the calculated tour is bounded by the size of the digraph times the shortest tour. Not only are the bounds worse for the directed case, but the running times of the approximating algorithms are also larger than those for the undirected case.INDEX TERMS--short tours, Hamiltonian circuits, sequential machines, knight's tour traveling salesman, approximating algorithms, NP-problem
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Generating a checking sequence with a minimum number of reset transitions
Given a finite state machine M, a checking sequence is an input sequence that is guaranteed to lead to a failure if the implementation under test is faulty and has no more states than M. There has been much interest in the automated generation of a short checking sequence from a finite state machine. However, such sequences can contain reset transitions whose use can adversely affect both the cost of applying the checking sequence and the effectiveness of the checking sequence. Thus, we sometimes want a checking sequence with a minimum number of reset transitions rather than a shortest checking sequence. This paper describes a new algorithm for generating a checking sequence, based on a distinguishing sequence, that minimises the number of reset transitions used.This work was supported in part by Leverhulme Trust grant number F/00275/D, Testing State Based Systems, Natural Sciences and Engineering Research Council (NSERC) of Canada grant number RGPIN 976, and Engineering and Physical Sciences Research Council grant number GR/R43150, Formal Methods and Testing (FORTEST)
Incremental Dead State Detection in Logarithmic Time
Identifying live and dead states in an abstract transition system is a
recurring problem in formal verification; for example, it arises in our recent
work on efficiently deciding regex constraints in SMT. However,
state-of-the-art graph algorithms for maintaining reachability information
incrementally (that is, as states are visited and before the entire state space
is explored) assume that new edges can be added from any state at any time,
whereas in many applications, outgoing edges are added from each state as it is
explored. To formalize the latter situation, we propose guided incremental
digraphs (GIDs), incremental graphs which support labeling closed states
(states which will not receive further outgoing edges). Our main result is that
dead state detection in GIDs is solvable in amortized time per edge
for edges, improving upon per edge due to Bender, Fineman,
Gilbert, and Tarjan (BFGT) for general incremental directed graphs.
We introduce two algorithms for GIDs: one establishing the logarithmic time
bound, and a second algorithm to explore a lazy heuristics-based approach. To
enable an apples-to-apples experimental comparison, we implemented both
algorithms, two simpler baselines, and the state-of-the-art BFGT baseline using
a common directed graph interface in Rust. Our evaluation shows -x
speedups over BFGT for the largest input graphs over a range of graph classes,
random graphs, and graphs arising from regex benchmarks.Comment: 22 pages + reference
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Optimizing the length of checking sequences
A checking sequence, generated from a finite state machine, is a test sequence that is guaranteed to lead to a failure if the system under test is faulty and has no more states than the specification. The problem of generating a checking sequence for a finite state machine M is simplified if M has a distinguishing sequence: an input sequence D~ with the property that the output sequence produced by M in response to D is different for the different states of M. Previous work has shown that, where a distinguishing sequence is known, an efficient checking sequence can be produced from the elements of a set A of sequences that verify the distinguishing sequence used and the elements of a set /spl gamma/ of subsequences that test the individual transitions by following each transition t by the distinguishing sequence that verifies the final state of t. In this previous work, A is a predefined set and /spl gamma/ is defined in terms of A. The checking sequence is produced by connecting the elements of /spl gamma/ and A to form a single sequence, using a predefined acyclic set E/sub c/ of transitions. An optimization algorithm is used in order to produce the shortest such checking sequence that can be generated on the basis of the given A and E/sub c/. However, this previous work did not state how the sets A and E/sub c/ should be chosen. This paper investigates the problem of finding appropriate A and E/sub c/ to be used in checking sequence generation. We show how a set A may be chosen so that it minimizes the sum of the lengths of the sequences to be combined. Further, we show that the optimization step, in the checking sequence generation algorithm, may be adapted so that it generates the optimal E/sub c/. Experiments are used to evaluate the proposed method
Testing from a finite state machine: Extending invertibility to sequences
When testing a system modelled as a finite state machine it is desirable to minimize the effort required. It has been demonstrated that it is possible to utilize test sequence overlap in order to reduce the test effort and this overlap has been represented by using invertible transitions. In this paper invertibility will be extended to sequences in order to reduce the test effort further and encapsulate a more general type of test sequence overlap. It will also be shown that certain properties of invertible sequences can be used in the generation of state identification sequences
Global approaches to solving recognition problems of noisy images, 1989
An important problem in the area of pattern recognition is automatic detection of certain pre-assigned elements of an image distorted by noise. In this research, a global ap proach will be used. One such approach is to use an optimal smoothing algorithm which depends on efficient dynamic programming computational techniques. The basic purpose of this research is to make this dynamic programming process efficient in terms of storage requirement and computational effort. Our goal, using the objective function, is to find an optimal order of optimization and then design an effi cient computational technique. Two global techniques will be presented in this paper. Included is a graph-searching technique and the above men tioned technique using dynamic programming. Emphasis will be on the development of an algorithm using dynamic program ming. I wish to express ray deepest appreciation and sincere gratitude to those who have contributed their time, energy, and support to make this study possible. Thanks are especially due Dr. Warsi, Nazir A. , my thesis advisor. His instruction, suggestions, and patience were essential to the completion of this thesi s. Further thanks are also due Nasa Langley for providing financial, technical, and general support to help make this study possible. Special thanks are offered to Mr. Micheal Goode, Technical Monitor, for his technical support and to Dr. Samuel E. Massenberg, University Affairs Officer, for his general support
SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning
We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a
node-disjoint cycle cover in a directed graph with minimum interaction cost
between successive arcs. We derive several semidefinite programming (SDP)
relaxations and use facial reduction to make these strictly feasible. We
investigate a nontrivial relationship between the transformation matrix used in
the reduction and the structure of the graph, which is exploited in an
efficient algorithm that constructs this matrix for any instance of the
problem. To solve our relaxations, we propose an algorithm that incorporates an
augmented Lagrangian method into a cutting plane framework by utilizing
Dykstra's projection algorithm. Our algorithm is suitable for solving SDP
relaxations with a large number of cutting planes. Computational results show
that our SDP bounds and our efficient cutting plane algorithm outperform other
QCCP bounding approaches from the literature. Finally, we provide several
SDP-based upper bounding techniques, among which a sequential Q-learning method
that exploits a solution of our SDP relaxation within a reinforcement learning
environment
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Dynamics of neural nets
In this short paper, I plan to review the work I have done on neural nets over the course of the last 20 years. As one might reasonably expect the questions being asked and the approaches to solving them have evolved, but there are still fundamental questions which remain unanswered and are perhaps unanswerable with present techniques.
I have used the word "dynamics" in the title to indicate that the major emphasis of the paper will be how the state of a neural net changes in the course of time either autonomously, that is without input, or in response to input. The problems of learning in neural nets will not be directly addressed, although I will argue that many questions about learning can be recast as questions about dynamics
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