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- DISCRETE APPLIED MATHEMATICS, SPECIAL ISSUE ON COMBINATORICS OF SEARCHING, SORTING, AND CODING. (2002) , 2004
"... We investigate a problem from computational biology: Given a constant size alphabet M with a weight function / : M--> +, find an efficient data structure and query algorithm solving the following problem: For a weight M C + and a string cr over A, decide whether cr contains a substring with weigh ..."
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We investigate a problem from computational biology: Given a constant size alphabet M with a weight function / : M--> +, find an efficient data structure and query algorithm solving the following problem: For a weight M C + and a string cr over A, decide whether cr 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 _ 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(,) space and (Wg ' loglog,) time; INTERVAL solves the problem for binary alphabets with O0, ) space in O(log,) time. We sketch a third al-gorithm, 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.