Efficient algorithm to sort linear combinations of arrays

Abstract

There have been many sorting algorithms that sort a given array of n numbers. In Computational Geometry it is often required to sort different arrays which are essentially the linear combinations of a fixed set of arrays. In this paper we claim that one can presort an array of ordered pairs for all possible combinations of the elements in each pair. Thus it is not necessary to employ any of the existing sorting algorithm for each of the linear combinations separately. On the other hand, the information can be globally compiled for the given pair of arrays in advance and actual sorting can be accomplished by employing a binary search. Thus any array resulting from the linear combination of an array of ordered pairs can be sorted in O(log n). The presorting is done by well known angle sweep paradigm and we prove certain results, which are otherwise trivially accepted regarding this paradigm. As a result we show here that angle-sweep technique can be implemented more efficiently. Moreover, since this process generates $\frac{{n\left( {n - 1} \right)}}{2}$ permutations out of all n! permutations, whether a given arrangement of points is possible can be determined in O(n ²) time. We show that these observations can lead to very simple techniques to solve certain well-known problem regarding arrangements of planar points.