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# Query-Competitive Sorting with Uncertainty

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LIPIcs.MFCS.2019.7.pdf
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## Cite As

Magnús M. Halldórsson and Murilo Santos de Lima. Query-Competitive Sorting with Uncertainty. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.7

## Abstract

We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries. We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2. We then present a unified adaptive strategy for uniform query costs that yields: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio 3/2 + O(1/k) if the components obtained have size at least k; (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components. We also show that the advice complexity of the adaptive problem is floor[n/2] if no error threshold is allowed, and ceil[n/3 * lg 3] for the general case.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Mathematics of computing → Discrete mathematics
• Theory of computation → Theory and algorithms for application domains
##### Keywords
• online algorithms
• sorting
• randomized algorithms
• threshold tolerance graphs

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