  Let $G$ be an edge-weighted directed graph with $n$ vertices embedded on an
orientable surface of genus $g$. We describe a simple deterministic
lexicographic perturbation scheme that guarantees uniqueness of minimum-cost
flows and shortest paths in $G$. The perturbations take $O(gn)$ time to
compute. We use our perturbation scheme in a black box manner to derive a
deterministic $O(n \log \log n)$ time algorithm for minimum cut in
\emph{directed} edge-weighted planar graphs and a deterministic $O(g^2 n \log
n)$ time proprocessing scheme for the multiple-source shortest paths problem of
computing a shortest path oracle for all vertices lying on a common face of a
surface embedded graph. The latter result yields faster deterministic
near-linear time algorithms for a variety of problems in constant genus surface
embedded graphs.
  Finally, we open the black box in order to generalize a recent linear-time
algorithm for multiple-source shortest paths in unweighted undirected planar
graphs to work in arbitrary orientable surfaces. Our algorithm runs in $O(g^2 n
\log g)$ time in this setting, and it can be used to give improved linear time
algorithms for several problems in unweighted undirected surface embedded
graphs of constant genus including the computation of minimum cuts, shortest
topologically non-trivial cycles, and minimum homology bases.
