For an undirected n-vertex graph G with non-negative edge-weights, we
consider the following type of query: given two vertices s and t in G,
what is the weight of a minimum st-cut in G? We solve this problem in
preprocessing time O(nlog3n) for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space O(n) that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
2O(g2)