  The round-trip distance function on a geographic network (such as a road
network, flight network, or utility distribution grid) defines the "distance"
from a single vertex to a pair of vertices as the minimum length tour visiting
all three vertices and ending at the starting vertex. Given a geographic
network and a subset of its vertices called "sites" (for example a road network
with a list of grocery stores), a two-site round-trip Voronoi diagram labels
each vertex in the network with the pair of sites that minimizes the round-trip
distance from that vertex. Alternatively, given a geographic network and two
sets of sites of different types (for example grocery stores and coffee shops),
a two-color round-trip Voronoi diagram labels each vertex with the pair of
sites of different types minimizing the round-trip distance. In this paper, we
prove several new properties of two-site and two-color round-trip Voronoi
diagrams in a geographic network, including a relationship between the
"doubling density" of sites and an upper bound on the number of non-empty
Voronoi regions. We show how those lemmas can be used in new algorithms
asymptotically more efficient than previous known algorithms when the networks
have reasonable distribution properties related to doubling density, and we
provide experimental data suggesting that road networks with standard
point-of-interest sites have these properties.
