Abstract This paper considers embeddings f of arbitrary finite metrics into the line metric! so thatnone of the distances is shrunk by the embedding f; the quantity of interest is the factor by whichthe average distance in the metric is stretched. We call this quantity the average distortion of the non-contracting map f.We prove that finding the best embedding of even a tree metric into a line metric so as to minimize the average distortion is NP-hard, and hence focus on approximating the average distortion of thebest possible embedding for the given input metric. We give a constant-factor approximation for the problem of embedding general metrics into the line metric. For the case of n-point tree metrics,we provide a quasi-polynomial time approximation scheme (QPTAS) which outputs an embedding with distortion at most (1 + ffl) times the optimum in time nO(log n/ffl 2). Finally, when the average

One of the fundamental issues that arises in computational biology is Multiple Sequence Alignment (MSA), which needs to be addressed in many applications of Bioinformatics (e.g. study of the SARS Coronavirus and the Human Genome Project). Many algorithms have been proposed to solve the MSA problem, but often cannot incorporate users' (biologists') knowledge of the functionalities or structures of these sequences into their solutions. This kind of information is very useful for an accurate and biologically meaningful alignment. The Constrained Multiple Sequence Alignment (CMSA) was proposed by Tang et al. (2002) to rectify the shortcomings of MSA by introducing a constrained sequence to represent more important residues in the sequences. Every character of the constrained sequence has to appear in an entire column in the alignment of the multiple sequences, and in the same order as in the constrained sequence.

Summary. The distance between two vertices is the basis of the definition of several graph parameters including diameter, radius, average distance and metric dimension. These invariants are examined, especially how they relate to one another and to other graph invariants and their behaviour in certain graph classes. We also discuss characterizations of graph classes described in terms of distance or shortest paths. Finally, generalizations are considered. 1 Overview of Chapter The distance between two vertices in a graph is a simple but surprisingly useful notion. It has led to the definition of several graph parameters such as the diameter, the radius, the average distance and the metric dimension. In this chapter we examine these invariants; how they relate to one another and other graph invariants and their behaviour in certain graph classes. We also discuss characterizations of graph classes that have properties that are described in terms of distance or shortest paths. We later consider generalizations of shortest paths connecting pairs of vertices to shortest trees, called