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

TL;DR

This paper determines the exact optimal gap sequences for Shellsort when sorting up to 16 elements and provides conjectures and best known sequences for 17 to 30 elements, revealing some sequences with large increments.

Contribution

It introduces a minimax search method in reduced permutational spaces to find exact optimal gap sequences for small n and lists best sequences with conjectures for larger n.

Findings

Exact optimal gap sequences for n ≤ 16 are established.

Best known gap sequences for 17 ≤ n ≤ 30 are listed with conjectures.

Some optimal sequences include increments larger than half of n.

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.