A tight analysis of the (1 + 1)-EA for the single source shortest path problem

We conduct a rigorous analysis of the (1 + 1) evolutionary algorithm for the single source shortest path problem proposed by Scharnow, Tinnefeld and Wegener (Journal of Mathematical Modelling and Algorithms, 2004). We prove a tight bound of Theta(n
²
max{log(n),lscr}) on the optimization time, where lscr is the maximum number of edges of a shortest path with minimum number of edges from the source to another vertex. Using various tools from probability theory we show that these bounds not only hold in expectation, but also with high probability. We are optimistic that these tools can also be used to analyze the run-time of evolutionary algorithms for other problems.