Sorting by Placement and Shift

TL;DR

This paper proves that a placement-based sorting algorithm terminates within a specific bound and characterizes permutations that reach this worst-case scenario, confirming a conjecture and revealing a super-exponential number of such permutations.

Contribution

It establishes the termination bound for the placement and shift sorting algorithm and confirms a conjecture, using a novel symmetrical binary representation.

Findings

Algorithm terminates after at most 2^{n-1}-1 steps in the worst case.

Super-exponentially many permutations reach the worst-case bound.

The proof introduces a unique symmetrical binary representation.

Abstract

In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of this algorithm is commonly used to hand-sort files.) However, it is not obvious that this algorithm necessarily terminates. We show that in fact the algorithm terminates after at most $2^{n-1}-1$ steps in the worst case (confirming a conjecture of L. Larson), and that there are super-exponentially many permutations for which this exact bound can be achieved. The proof involves a curious symmetrical binary representation.