Tight Bounds for Online Stable Sorting

TL;DR

This paper establishes tight bounds for online stable sorting, showing that the number of comparisons needed closely matches the entropy-based lower and upper bounds, thus advancing understanding of sorting efficiency.

Contribution

It provides the first bounds that are within o(n) of each other for online stable sorting, aligning with offline sorting bounds, especially for small sigma.

Findings

For sigma = o(n / log n), (H + 1) n + o(n) comparisons suffice.

A matching lower bound of (H + 1) n - o(n) comparisons is proven.

Bounds are close to offline sorting bounds, improving theoretical understanding.

Abstract

Although many authors have considered how many ternary comparisons it takes to sort a multiset $S$ of size $n$, the best known upper and lower bounds still differ by a term linear in $n$. In this paper we restrict our attention to online stable sorting and prove upper and lower bounds that are within (o (n)) not only of each other but also of the best known upper bound for offline sorting. Specifically, we first prove that if the number of distinct elements (\sigma = o (n / \log n)), then ((H + 1) n + o (n)) comparisons are sufficient, where $H$ is the entropy of the distribution of the elements in $S$. We then give a simple proof that ((H + 1) n - o (n)) comparisons are necessary in the worst case.