Use of simple polygonal chains in generating random simple polygons

Abstract

The main motivation for generating random simple polygons is to produce test instances for geometric algorithms. In this paper three new algorithms are proposed to generate random simple polygons. A point set in a two dimensional plane is the input, and a simple polygon is the output of the problem. At first a new algorithm to convert any kind of simple polygonal chains into simple polygons is presented and the correctness of the algorithm is proved. Then three new algorithms are presented to produce random simple polygonal chains from the given point set. The first algorithm is capable of producing $2^n$ simple polygonal chains. The second algorithm works by the concept of divide and conquer and the third algorithm is the most complete and produces all the possible simple polygonal chains. The worst time complexities of these three chain generation algorithms are $O(n^2)$, $O(n^2)$ and $O(n^3)$ respectively and the time complexity of the conversion algorithm is O(n*l), where $1<l<n$. The polygon generation algorithm works in this way that first each simple polygonal chain generation algorithms are applied over the point set and then the generated chains are converted to simple polygons. The number of different simple polygons generated by each of three algorithms is compared with the well-known algorithms and the experimental results show that the third algorithm produces more polygons rather than the well-known 2-opt move algorithm. The first algorithm acts better than the second algorithm, where both act better than steady Growth.