Lower Bounds for Sorting 16, 17, and 18 Elements

TL;DR

This paper determines the exact minimum comparison counts for sorting 16, 17, and 18 elements, filling a long-standing gap and disproving a previous conjecture, using exhaustive computer search and novel algorithms.

Contribution

It provides the exact comparison bounds for sorting 16-18 elements, resolving open questions and introducing new algorithmic techniques for such problems.

Findings

S(16)=46 comparisons

S(17)=50 comparisons

S(18)=54 comparisons

Abstract

It is a long-standing open question to determine the minimum number of comparisons $S(n)$ that suffice to sort an array of $n$ elements. Indeed, before this work $S(n)$ has been known only for $n\leq 22$ with the exception for $n=16$, $17$, and $18$. In this work, we fill that gap by proving that sorting $n=16$, $17$, and $18$ elements requires $46$, $50$, and $54$ comparisons respectively. This fully determines $S(n)$ for these values and disproves a conjecture by Knuth that $S(16) = 45$. Moreover, we show that for sorting $28$ elements at least 99 comparisons are needed. We obtain our result via an exhaustive computer search which extends previous work by Wells (1965) and Peczarski (2002, 2004, 2007, 2012). Our progress is both based on advances in hardware and on novel algorithmic ideas such as applying a bidirectional search to this problem.