Partially supported by Icelandic Research Fund grant 174484-051. A preliminary version of this paper appeared in volume 138 of LIPICs, article 7, 2019. DOI: 10.4230/LIPIcs.MFCS.2019.7
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 poly time, and that both oblivious and adaptive problems have simple competitive algorithms. The competitive ratio for the oblivious problem is $n$ for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.
We present a unified adaptive strategy for uniform costs that yields the following improved results: (1) a 3/2-competitive randomized algorithm; (2) a 5/3-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has competitive ratio $3/2+\mathrm{O}(1/k)$ if the components obtained have size at least $k$; and (3) an exact algorithm for laminar families of intervals. 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 competitive ratio $1+\frac{4}{3\sqrt{3}}\approx 1.7698$ for arbitrary query costs, and we show that the 2-competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general vertex cover problem, by using the local ratio technique. Moreover, we prove that the advice complexity of the adaptive problem is $\lfloor n/2\rfloor$ if no error threshold is allowed, and $\lceil n/3\cdot\lg 3\rceil$ for the general case.
Finally, we present some graph-theoretical results on co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.
Sorting is a foundational problem in computer science that is typically employed on sequences or total orders. More recently, a more general form of sorting on partially ordered sets (or posets), where some pairs of elements are incomparable, has been studied. General poset sorting algorithms have a lower-bound query complexity of $Ω(wn + n \log n)$, where $w$ is the width of the poset. We consider the problem of sorting in trees, a particular case of partial orders, and parametrize the complexity with respect to $d$, the maximum degree of an element in the tree, as $d$ is usually much smaller…
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We study how to utilize (possibly machine-learned) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. The goal is to minimize the number of queries needed to solve the problem. We consider fundamental problems such as finding the minima of intersecting sets of elements or sorting them (these problems can also be phrased as (hyper)graph orientation problems), as well as the minimum spanning tree problem. We discuss different measures for the prediction accuracy and design algorithms with performance guarantees that improve with the accuracy of p…
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The area of computing with uncertainty considers problems where some information about the input elements is uncertain, but can be obtained using queries. For example, instead of the weight of an element, we may be given an interval that is guaranteed to contain the weight, and a query can be performed to reveal the weight. While previous work has considered models where queries are asked either sequentially (adaptive model) or all at once (non-adaptive model), and the goal is to minimize the number of queries that are needed to solve the given problem, we propose and study a new model where $…
We consider online algorithms for the $k$-server problem on trees. There is a $k$-competitive algorithm for this problem, and it is the best competitive ratio. M. Chrobak and L. Larmore provided it. At the same time, the existing implementation has $O(n)$ time complexity for processing a query and $O(n)$ for prepossessing, where $n$ is the number of nodes in a tree. Another implementation of the algorithm has $O(k^2+k\log n)$ time complexity for processing a query and $O(n\log n)$ for prepossessing. We provide a new time-efficient implementation of the algorithm. It has $O(n)$ time complexity …
