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Abstract: Circular strings are different from linear strings in that the last symbol is considered to precede the first symbol. Even though circular strings are biologically important, only a few efforts have been made to solve computational problems on circular strings. In this paper, we introduce consensus problems for circular strings of length and present the first non-trivial algorithms to find a consensus and an optimal alignment for circular strings by the Hamming distance. They are -time algorithms for three circular strings and an -time algorithm for four circular strings. Our algorithms are times faster than the naïve algorithms directly using the solutions for the linear consensus problems, which take time for three circular strings and time for four circular strings. This speedup was achieved by reducing the problems into correlations and by formulating and solving systems of linear equations. Moreover, our algorithms use only space. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an algorithm which solves the membership problem of Petri net controlled grammars without -rules and cyclic rules. We define a conditional tree which is a modified derivation tree of a context-free grammar with information about control by a Petri net. It is shown that a conditional tree is cancelled to a derivation tree without conditions if and only if there is a derivation under the control of the Petri net from the start symbol to a word which is the yielding of the conditional tree. Then the Earley’s algorithm is extended to make a conditional tree in addition to parse a word. Thus the word is generated by a given Petri net controlled grammar if and only if the resulting conditional tree is cancelled to a tree of no condition. The time complexity of the algorithm is nondeterministic polynomial of the length of an input word. Therefore the class of languages generated by Petri net controlled grammars without -rules and cyclic rules is included in the class of context-sensitive languages. [Copyright &y& Elsevier]

Abstract: This paper presents a new polynomial-time algorithm to solve the minimum cost tension problem. It runs in -time, where denote the number of nodes, number of arcs, and maximum arc capacity value of an arc cost, respectively. The algorithm improves the -time algorithm of Maurras (1994) . Also our algorithm, under the similarity assumption (Gabow, 1985) , runs in -time, which improves the -time algorithm of Hadjiat and Maurras (1997) . [Copyright &y& Elsevier]

Abstract: In this paper, we consider an online non-preemptive scheduling problem on two related machines, where at most jobs are allowed to be rearranged, but only after all jobs have been revealed and (temporarily) scheduled. We minimize the makespan, and we call the problem as Online scheduling with bounded rearrangement at the end (BRE), which is a semi-online problem. Jobs arrive one by one over list. After all the jobs have been arrived and scheduled, we are informed that the input sequence is over; then at most already scheduled jobs can be reassigned. With respect to the worst case ratio, we close the gap between the lower bound and upper bound, improving the previous result as well. Especially, for the lower bound, (i) for an improved lower bound is obtained, which is better than (Liu et al. (2009) ); (ii) for , an improved lower bound is obtained, which is better than (Liu et al. (2009) ). For the upper bound, (i) for and , a new upper bound is obtained, which is optimal and better than the one in Liu et al. (2009) ; (ii) for and , an upper bound is proposed, which is optimal and better than the previous one in Liu et al. (2009) ; (iii) for and , an upper bound is obtained, which is also optimal and better than the previous one in Liu et al. (2009) . [Copyright &y& Elsevier]