Algorithmic complexity of protein identification: combinatorics of weighted strings

Abstract We investigate a problem which arises in computational biology: Given a constant-size alphabet A with a weight function μ : A → N , find an efficient data structure and query algorithm solving the following problem: For a string σ over A and a weight M∈ N , decide whether σ contains a substring with weight M , where the weight of a string is the sum of the weights of its letters (O NE -S TRING M ASS F INDING P ROBLEM ). 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 n of the input string σ . Among others, we present two non-trivial efficient algorithms: L OOKUP solves the problem with O( n ) storage space and O (n/ log n) time; I NTERVAL solves the problem for binary alphabets with O( n ) storage space in O ( log n) query time. We introduce other variants of the problem and sketch how our algorithms may be extended for these variants. Finally, we discuss combinatorial properties of weighted strings.

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