Empty pseudo-triangles in point sets

On Counting and Analyzing Empty Pseudo-triangles in a Point Set

“…A year ago, Aichholzer et al [3] investigated non-convex 4-holes in a point set, which essentially represent non-convex quadrilaterals, and can be viewed as a special case of pseudo-triangles. Finally, very recently, Ahn et al [5] have further elaborated and extended the results from [10] and [4] and summarized them in a joint paper.…”

“…Of course, the method by Ahn et al [4,5] allows accomplishing this task; however, we demonstrate that the counting problem can be solved at a lower computational cost than the enumerating one. Moreover, our proposed approach extends to finding an empty pseudo-triangle with the maximum perimeter or the maximum area, resulting in a simpler and clearer algorithm than the one described by Ahn et al [5].…”

“…In [5], Ahn et al are looking, in particular, for an empty pseudo-triangle with the maximum perimeter or the maximum area. Thereby, their algorithms actually enumerate all the empty pseudo-triangles defined by P .…”

“…Two years later, Ahn et al [4] examined the problem of generating such pseudo-triangles optimal in a certain sense-and namely, those with the minimum perimeter, the maximum area, or the minimum longest convex chain. (As a matter of fact, Ahn et al [4] used a more general definition of a pseudo-triangle, allowing its corners to be connected by concave curves, and not necessarily polygonal chains. However, the respective optimization problems immediately reduce to those of finding an optimal polygonal pseudo-triangle, defined as above.)…”

“…We shall follow the scheme suggested by van Kreveld and Speckmann [10] and further adopted by Ahn et al [4,5], and first restrict our attention to the case when some three points from P are selected, and we are interested only in the empty pseudo-triangles with the corners at those points. Then the worstcase number of general and of star-shaped empty pseudo-triangles is known to be Θ(n 3 ) and Θ(n 2 ), respectively [10], and the method by Ahn et al [4,5] adapted for solving the corresponding counting problems will require O(n 3 ) time and O(n) space in either case.…”