| Abstract: |
We investigate a problem from computational biology: Given a constant-size alphabet A with a weight function m mapping each letter to a positive real number, find an efficient data structure
and query algorithm solving the following problem: For a weight M and a string s over A, decide whether s contains a substring with weight M (One-String Mass Finding Problem). If the answer is
yes, then we may in addition require a witness, i.e. indices i, j such that the substring beginning at position i and ending at position j has weight M. We allow preprocessing of the string, and
measure efficiency in two parameters: storage space required for the preprocessed data, and running time of the query algorithm for given M. We are interested in data structures and algorithms
requiring subquadratic storage space and sublinear query time, where we measure the input size as the length of the input string. We present two efficient algorithms: Lookup solves the problem
with O(n) space and O(n*log(log(n))/log n) time; Interval solves the problem for binary alphabets with O(n) space in O(log n) time. We sketch a third algorithm, Cluster, which can be adjusted for
a space-time-tradeoff but for which we do not yet have a resource analysis. We introduce a function on weighted strings which is closely related to the analysis of algorithms for the One-String
Mass Finding Problem: The number of different submasses of a weighted string. We present several properties of this function, including upper and lower bounds. Finally, we introduce two more
general variants of the problem and sketch how algorithms may be extended for these variants. |
