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 the following improved results: (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; and (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 give a randomized adaptive algorithm with query-competitive factor 1 + 4 3 √ 3 ≈ 1.7698 for arbitrary query costs, and we show that the 2-query competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general graph problem (which is also a generalization of the vertex cover problem), by using the local ratio technique. Furthermore, we prove 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.Finally, we present some graph-theoretical results regarding co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.

Round-Competitive Algorithms for Uncertainty Problems with Parallel Queries

“…This is equivalent to saying that I i and I j are dependent if and only if u i > ℓ j and u j > ℓ i . Two simple facts are important to notice, which are proven in [21]:…”

“…, by performing the minimum number of queries possible. This problem was addressed for k = 1 in [21,22,27]; it admits 2-query-competitive deterministic algorithms and has a deterministic lower bound of 2.…”

“…We adapt the 2-query-competitive algorithm for Sorting by Halldórsson and de Lima [21] for k = 1 to the case of arbitrary k. Their algorithm first queries all non-trivial intervals in a minimum vertex cover in the dependency graph. By the duality between vertex covers and independent sets, the unqueried intervals form an independent set, so no query is necessary to decide the order between them.…”

“…For the Selection problem, we obtain a 2-round-competitive algorithm. For Sorting, we show that there is a 2-round-competitive algorithm, by adapting ideas from a recent algorithm for sorting in the standard adaptive model [21], and that this is best possible.…”

“…The problem of sorting uncertainty data has received some attention recently. Halldórsson and de Lima [21] presented better query-competitive algorithms, by using randomization or assumptions on the underlying graph structure. Other related work on sorting has considered sorting with noisy information [1,6] or preprocessing the uncertain intervals so that the actual numbers can be sorted efficiently once their precise value are revealed [31].…”