eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-01-24
56:1
56:17
10.4230/LIPIcs.ITCS.2024.56
article
An Algorithm for Bichromatic Sorting with Polylog Competitive Ratio
Goswami, Mayank
1
https://orcid.org/0000-0002-2111-3210
Jacob, Riko
2
https://orcid.org/0000-0001-9470-1809
Queens College CUNY, Flushing, New York, NY, USA
IT University of Copenhagen, Denmark
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol287-itcs2024/LIPIcs.ITCS.2024.56/LIPIcs.ITCS.2024.56.pdf
Sorting
Priced Information
Nuts and Bolts