Acceleration of sweep-line technique by employing smart quicksort

Quicksort is usually the best practical choice for sorting because it is, on average, remarkably efficient. Unfortunately, this popular algorithm has a significant drawback: the slowest performance is obtained in the simplest cases when input data are already initially sorted or only a slight perturbation occurs. In this paper, we propose a combination of quicksort and a new algorithm, which shows excellent time performance in sorting such crucial data arrays, and which is not much slower than quicksort in random cases. Our work was inspired by problems met when sorting polygon vertices in the sweep-line algorithms of computational geometry and, therefore, we have named the new algorithm 'vertex sort'. It splits the input array into three sub-arrays. Two of them are already sorted, and the third one is handled iteratively. A simple test decides whether to continue recursively with vertex sort or to employ quicksort in the second iteration. In this way, we achieve a situation where the worst case time complexity does not exceed the running times of quicksort, but the simplest cases are handled much faster (in linear time) than random cases. We have named the combined algorithm 'smart quicksort' because of this desired property. In the last part of the paper, we prove its efficiency by employing it in a well-known sweep-line-based polygon triangulation algorithm.