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In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time n O (k) . This improves the naive algorithm that runs in time n O (k 2) by Espelage et al. (Graph-theoretic concepts in computer science, vol 2204. Springer, Berlin, 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time n o (k) (Fomin et al. in SIAM J Comput 43:1541–1563, 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges. [ABSTRACT FROM AUTHOR]

In 1994, Klapper and Goresky (Proceedings of the 1993 Cambridge Security Workshop, Lecture Notes in Computer Science, vol 809, Cambridge, pp 174–178, 1994) proposed a new device called feedback with carry shift register to generate pseudo-random sequences instead of using the traditional device linear feedback shift register. They raised an algorithm called as rational approximation algorithm to recover the device for a given sequence (Klapper and Goresky, Advances in Cryptology, Crypto'95, Lecture Notes in Computer Science, vol 963, Springer, Berlin, pp 262–274, 1995). In this paper, we propose a new algorithm by introducing a new parameter and get the best rational approximation of the sequence much more quickly, especially when the size of the sequence increases dramatically. Unlike most of known algorithms, we can solve the minimal lattice basis instead of one shortest vector. Besides, we can prove that the solution of each step is optimal regardless of the length of the input sequence theoretically. [ABSTRACT FROM AUTHOR]