Color Set Size problem with applications to string matching

The Color Set Size problem is: Given a rooted tree of size n with l leaves colored from 1 to m, m ≤ l, for each vertex u find the number of different leaf colors in the subtree rooted at u. This problem formulation, together with the Generalized Suffix Tree data structure has applications to string matching. This paper gives an optimal sequential solution of the color set size problem and string matching applications including a linear time algorithm for the problem of finding the longest substring common to at least k out of m input strings for all k between 1 and m. In addition, parallel solutions to the above problems are given. These solutions may shed light on problems in computational biology, such as the multiple string alignment problem.