Sorting using non-binary comparisons

TL;DR

This paper explores a generalized sorting problem using a scale that compares multiple coins at once, providing optimal algorithms for both online and offline scenarios with non-binary comparisons.

Contribution

It introduces a novel multi-input comparison model for sorting and develops optimal algorithms for both online and offline settings.

Findings

Established lower bounds for query complexity.

Designed optimal algorithms matching these bounds.

Extended classical sorting to non-binary comparison models.

Abstract

In this paper we investigate the problem of sorting a set of $n$ coins, each with distinct but unknown weights, using an unusual scale. The classical version of this problem, which has been well-studied, gives the user a binary scale, enabling them to determine which is the lighter/heavier of any two objects. We generalise this, considering a scale that accepts $k$ coins as input and returns the $t^{\text{th}}$ lightest, for a fixed $k$ and $t$. We consider this in both an on-line and off-line setting, and exhibit algorithms in both settings that are best-possible in terms of the order of the number of queries required.