  We present an algorithm for efficiently exploring inequivalent Calabi-Yau
threefold hypersurfaces in toric varieties. A direct enumeration of fine,
regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke
database is foreseeably impossible due to the large count of distinct FRSTs.
Moreover, such an enumeration is needlessly redundant because many such
triangulations have the same restrictions to 2-faces and hence, by Wall's
theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy
can be circumvented by finding a height vector in the strict interior of the
intersection of the secondary cones associated with each 2-face triangulation.
We demonstrate that such triangulations are generated with orders of magnitude
fewer operations than the naive approach of generating all FRSTs and selecting
only those differing on 2-faces. Similar methods are also presented to directly
generate (the support of) the secondary subfan of all fine triangulations,
relevant for random sampling of FRSTs.
