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# An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio

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LIPIcs.ITCS.2024.56.pdf
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## Acknowledgements

We want to thank an anonymous referee for pointing out another natural variant of bichromatic sorting.

## Cite As

Mayank Goswami and Riko Jacob. An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.56

## Abstract

The problem of sorting with priced information was introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to find a sorting algorithm with small competitive ratio, defined as the (worst-case) ratio of the algorithm’s cost to the cost of the cheapest proof of the sorted order. The simple case of bichromatic sorting posed by [CFGKRS] remains open: We are given two sets A and B of total size N, and the cost of an A-A comparison or a B-B comparison is higher than an A-B comparison. The goal is to sort A ∪ B. An Ω(log N) lower bound on competitive ratio follows from unit-cost sorting. Note that this is a generalization of the famous nuts and bolts problem, where A-A and B-B comparisons have infinite cost, and elements of A and B are guaranteed to alternate in the final sorted order. In this paper we give a randomized algorithm InversionSort with an almost-optimal w.h.p. competitive ratio of O(log³ N). This is the first algorithm for bichromatic sorting with a o(N) competitive ratio.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Abstract machines
• Theory of computation → Sorting and searching
##### Keywords
• Sorting
• Priced Information
• Nuts and Bolts

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