 Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations . This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
<clarity>  Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations . This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
 Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations , leading to an algorithm to compute such a minimum flip path in O\left({2}}}\right) .