 However, when restricted to lattice triangulations- i. e. triangulations of a finite subset of the integer lattice, or an affine transformation of this lattice, bounded by a simple, closed polygon with lattice points as vertices - the problem has a polynomial time algorithm. Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right)  .
<meaning-changed>  However, when restricted to lattice triangulations- i. e. triangulations of a finite subset of the integer lattice, or an affine transformation of this lattice, bounded by a simple, closed polygon with lattice points as vertices - the problem has a polynomial time algorithm. Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science\left(\right)  .
 However, for minimum flip paths between lattice triangulations, Eppstein and Caputo et al. gave algorithms running in O\left(\right) .