Finite state automata (FSA) are ubiquitous in computer science. Two of the most important algorithms for FSA processing are the conversion of a non-deterministic finite automaton (NFA) to a deterministic finite automaton (DFA), and then the production of the unique minimal DFA for the original NFA. We exhibit a parallel disk-based algorithm that uses a cluster of 29 commodity computers to produce an intermediate DFA with almost two billion states and then continues by producing the corresponding unique minimal DFA with less than 800,000 states. The largest previous such computation in the literature was carried out on a 512-processor CM-5 supercomputer in 1996. That computation produced an intermediate DFA with 525,000 states and an unreported number of states for the corresponding minimal DFA. The work is used to provide strong experimental evidence satisfying a conjecture on a series of token passing networks. The conjecture concerns stack sortable permutations for a finite stack and a 3-buffer. The origins of this problem lie in the work on restricted permutations begun by Knuth and Tarjan in the late 1960s. The parallel disk-based computation is also compared with both a single-threaded and multi-threaded RAM-based implementation using a 16-core 128 GB large shared memory computer.
Obtaining a minimal canonical DFA equivalent to a given NFA is important for the analysis of the classes of permutations generated by token passing in graphs. Such a graph is called a token passing network (TPN) @cite_9 @cite_3 . This is related to the subject permutations @cite_15 , with origins in the 1969 work of Knuth [Section 2.2.1] Knuth:1969 and the 1972 work of Tarjan @cite_23 . TPNs are used to model or approximate a range of data structures, including combinations of stacks, and provide tools for analyzing the classes of permutations that can be sorted or generated using them. Stack sorting problems have been the subject of extensive research @cite_35 . Sorting with two ordered stacks in series is detailed in @cite_36 . Permutation classes defined by TPNs are described in @cite_33 . Very recent work focused on permutations generated by stacks and dequeues @cite_21 . A collection of results on permutation problems expressed as token passing networks is in @cite_0 .
@cite_35 @cite_33 @cite_36 @cite_9 @cite_21 @cite_3 @cite_0 @cite_23 @cite_15
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Abstract A transportation graph is a directed graph with a designated input node and a designated output node. Initially, the input node contains an ordered set of tokens 1,2,3, … The tokens are removed from the input node in this order and transferred through the graph to the output node in a series of moves; each move transfers a token from a node to an adjacent node. Two or more tokens cannot reside on an internal node simultaneously. When the tokens arrive at the output node they will appear in a permutation of their original order. The main result is a description of the possible arrival permutations in terms of regular sets. This description allows the number of arrival permutations of each length to be computed. The theory is then applied to packet-switching networks and has implications for the resequencing problem. It is also applied to some complex data structures and extends previously known results to the case that the data structures are of bounded capacity. A by-product of this investigation is a new proof that permutations which avoid the pattern 321 are in one to one correspondence with those that avoid 312. This article proposes an enhanced differential fault analysis (DFA) method named as fault-propagation pattern-based DFA (FPP-DFA). The main idea of FPP-DFA is using the FPP of the ciphertext difference to predict the fault location and the fault-propagation path. It shows that FPP-DFA is very effective on SPN structure block ciphers using bitwise permutation, which is applied to two block ciphers. The first is PRESENT with the substitution-permutation sequence. With the fault model of injecting one nibble fault into the r-2nd round, on average 8 and 16 faults can reduce the key search space of PRESENT-80 128 to 214.7 and 221.1, respectively. The second is PRINTcipher with the permutation-substitution sequence. For the first time, it shows that although the permutation of PRINTcipher is secret key dependent, FPP-DFA still works well on it. With the fault model of injecting one nibble fault into the r-2nd round, 12 and 24 effective faults can reduce the key search space of PRINTcipher-48 96 to 213.7 and 222.8, respectively. A Lagrangian relaxation network for graph matching is presented. The problem is formulated as follows: given graphs G and g, find a permutation matrix M that brings the two sets of vertices into correspondence. Permutation matrix constraints are formulated in the framework of deterministic annealing. Our approach is in the same spirit as a Lagrangian decomposition approach in that the row and column constraints are satisfied separately with a Lagrange multiplier used to equate the two "solutions". Due to the unavoidable symmetries in graph isomorphism (resulting in multiple global minima), we add a symmetry-breaking self-amplification term in order to obtain a permutation matrix. With the application of a fixpoint preserving algebraic transformation to both the distance measure and self-amplification terms, we obtain a Lagrangian relaxation network. The network performs minimization with respect to the Lagrange parameters and maximization with respect to the permutation matrix variables. Simulation results are shown on 100 node random graphs and for a wide range of connectivities. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail.