Optimal Gap Sequences in Shellsort for $n\leq16$ Elements

Abstract

Optimal gap sequences in Shellsort, defined as gap sequences having the minimised maximum number of comparisons for a fixed number of pairwise distinct elements, are found by minimax search in reduced permutational spaces, namely Bad $(s,1)$-sorted permutations. Exact optimal gap sequences in Shellsort for $n\leq16$ pairwise distinct elements are established, and the best known gap sequences for $17\leq n\leq 30$ are listed with conjectures made. It notably discovers some optimal gap sequences consist of increments larger than the half of the total number of the elements to sort.