Query-competitive sorting with uncertainty

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 3/2query-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 n/2 if no error threshold is allowed, and n/3 • lg 3 for the general case.

Algorithms that Access the Input via Queries

“…For that special case we show that distributing the queries in a balanced way among all unsolved sets yields an O(log min{k, m}/ log log min{k, m})-roundcompetitive algorithm. We also give 2-round competitive algorithms for sorting multiple sets (by adapting ideas from previous work on sorting with uncertainty [12,19]) and for determining the i-th smallest value (and all elements whose value is equal to that value) of a single set.…”

Explorable Uncertainty Meets Decision-Making in Logistics

“…The line of research on explorable uncertainty has been initiated by Kahan (1991) in the context of selection problems. Subsequent work addressed caching problems (Olston and Widom 2000), problems such as computing a function value (Khanna and Tan 2001), finding the kth smallest value in a set of uncertainty intervals (Feder et al 2003;Gupta et al 2016), also with non-uniform query cost (Feder et al 2003), and sorting (Halldórsson and de Lima 2019).…”

“…Given an instance of the sorting problem, we can create a set for each pair of elements that are in the same set of the sorting instance and obtain a minimum problem whose feasible query sets also solve the sorting problem. Halldórsson and de Lima (2019) showed directly that the witness set algorithm for sorting a single set is 2-competitive and is best possible. They also show that the competitive ratio can be improved to 1.5 using randomization.…”