MPI-I-2003-1-018

Banderier, Beier and Mehlhorn showed that the single-source shortest path problem has smoothed complexity $O(m+n(K-k))$ if the edge costs are $K$-bit integers and the last $k$ least significant bits are perturbed randomly. Their analysis holds if each bit is set to $0$ or $1$ with probability $\frac{1}{2}$. We extend their result and show that the same analysis goes through for a large class of probability distributions: We prove a smoothed complexity of $O(m+n(K-k))$ if the last $k$ bits of each edge cost are replaced by some random number chosen from $[0, \dots, 2^k-1]$ according to some \emph{arbitrary} probability distribution $\pdist$ whose expectation is not too close to zero. We do not require that the edge costs are perturbed independently. The same time bound holds even if the random perturbations are heterogeneous. If $k=K$ our analysis implies a linear average case running time for various probability distributions. We also show that the running time is $O(m+n(K-k))$ with high probability if the random replacements are chosen independently.