Optimal recombination algorithms for generalized chains

The problem of sequence recombination has important applications in computational molecular biology. Recently, a distance problem involving recombination consisting of a single crossover was proposed. The problem is to generate a given set of sequences 𝒜 from a given pair of sequences 𝒮 in a minimum number of recombination (i.e., crossover) operations. For an arbitrary class 𝒜 of sequences, the computational complexity of the problem has not yet been settled so it is interesting to find polynomially solvable cases for this kind of recombination distance problem. In recent papers [Y. He and T. Chen, Optimal algorithms for recombination distance problem, Optim. Methods and Soft. 18 (2003), pp. 647–655], [S. Wu and X. Gu, A greedy algorithm for optimal recombination. Proceedings of COCOON2001, Lecture Notes in Computer Science 2108, 2001, 86–90], special classes of sequences called ‘tree’ and ‘chain’ were proposed, and polynomial time algorithms were designed for optimally generating tree and chain classes, respectively. In this paper, we define a new class 𝒜 of sequences, called generalized chain, which greatly extends the definition of chain class by ignoring some restrictive conditions. We distinguish a generalized chain into three cases: continuous chain, discontinuous chain and mixed chain. Then we present optimal algorithms for dealing with all cases of a generalized chain. All algorithms run in polynomial time.