On the Expressivity of Alignment-Based Distance and Similarity Measures on Sequences and Trees in Inducing Orderings

Abstract

Both ‘distance’ and ‘similarity’ measures have been proposed for the comparison of sequences and for the comparison of trees, based on scoring mappings. For a given alphabet of node-labels, the measures are parameterised by a table giving label-dependent values for swaps, deletions and insertions. The paper addresses the question whether an ordering by a ‘distance’ measure, with some parameter setting, can be also expressed by a ‘similarity’ measure, with some other parameter setting, and vice versa. Ordering of three kinds is considered: alignment-orderings, for fixed source S and target T, neighbour-orderings, where for a fixed S, varying candidate neighbours T _i are ranked, and pair-orderings, where for varying S _i , and varying T _j , the pairings $\langle {S}_{i},{T}_{j}\rangle$ are ranked. We show that (1) any alignment-ordering expressed by ‘distance’ setting be re-expressed by a ‘similarity’ setting, and vice versa; (2) any neigbour-ordering and pair-ordering expressed by a ‘distance’ setting be re-expressed by a ‘similarity’ setting; (3) there are neighbour-orderings and pair-orderings expressed by a ‘similarity’ setting which cannot be expressed by a ‘similarity’ setting. A consequence of this is that there are categorisation and hierarchical clustering outcomes which can be achieved via similarity but not via