Flip distance
flip distance in triangulations
In discrete mathematics and theoretical computer science, the flip distance between two triangulations of the same point set is the number of flips required to transform one triangulation into another. A flip removes an edge between two triangles in the triangulation and then adds the other diagonal in the edge's enclosing quadrilateral, forming a different triangulation of the same point set.
This problem is known to be NP-hard, even in its special case of determining the flip distance between convex polygons. Computing the flip distance between convex polygon triangulations is also equivalent to rotation distance, the number of rotations required to transform one binary tree into another.
Definition
Given a family of triangulations of some geometric object, a flip is an operation that transforms one triangulation to another by removing an edge between two triangles and adding the opposite diagonal to the resulting quadrilateral. The flip distance between two triangulations is the minimum number of flips needed to transform one triangulation into another. It can also be described as the shortest path distance in a flip graph, a graph that has a vertex for each triangulation and an edge for each flip between two triangulations. Flips and flip distances can be defined in this way for several different kinds of triangulations, including triangulations of sets of points in the Euclidean plane, triangulations of polygons, and triangulations of abstract manifolds.
Feasibility
The flip distance is well-defined only if any triangulation can be converted to any other triangulation via a sequence of flips. An equivalent condition is that the flip graph must be connected.
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