Lower Bounds for Sorting 16, 17, and 18 Elements
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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≤ 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.
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