Runtime analysis of the (1+1) evolutionary algorithm on strings over finite alphabets

In this work, we investigate a (1+1) Evolutionary Algorithm for optimizing functions over the space {0,...,<i>r</i>} ^<i>n</i>, where <i>r</i> is a positive integer. We show that for linear functions over {0,1,2}^<i>n</i>, the expected runtime time of this algorithm is O(<i>n</i> log <i>n</i>). This result generalizes an existing result on pseudo-Boolean functions and is derived using drift analysis. We also show that for large values of <i>r</i>, no upper bound for the runtime of the (1+1) Evolutionary Algorithm for linear function on {0,...,<i>r</i>}^<i>n</i> can be obtained with this approach nor with any other approach based on drift analysis with weight-independent linear potential functions.