Towards Optimal Sorting of 16 Elements

TL;DR

This paper investigates the minimal number of comparisons needed to sort 16 elements, using exhaustive computer search and an efficient counting algorithm, to resolve an open problem in sorting theory.

Contribution

It introduces an exhaustive search approach and a new algorithm for counting linear extensions to determine the optimal comparison count for 16 elements.

Findings

46 comparisons suffice to sort 16 elements

44 comparisons are insufficient for sorting 16 elements

The paper advances understanding of comparison bounds for small sorting problems

Abstract

One of the fundamental problem in the theory of sorting is to find the pessimistic number of comparisons sufficient to sort a given number of elements. Currently 16 is the lowest number of elements for which we do not know the exact value. We know that 46 comparisons suffices and that 44 do not. There is an open question if 45 comparisons are sufficient. We present an attempt to resolve that problem by performing an exhaustive computer search. We also present an algorithm for counting linear extensions which substantially speeds up computations.