Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments

Focke, Jacob; Megow, Nicole; Meißner, Julie

doi:10.4230/lipics.sea.2017.22

Cited by 5 publications

(8 citation statements)

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“…Furthermore, they showed that an optimum query set and the actual value of the minimum spanning tree can be computed in polynomial time. Some experimental evaluation of those algorithms were presented in [18].…”

Section: Related Workmentioning

confidence: 99%

Query-Competitive Sorting with Uncertainty

Halldórsson

Lima

2019

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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.

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Section: Related Workmentioning

confidence: 99%

Query-Competitive Sorting with Uncertainty

Halldórsson

Lima

2019

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“…Motivated by real-world applications where only rough information about the input data is initially available but precise information can be obtained at a cost, researchers have considered a range of uncertainty problems with queries [7,13,14,15,16,19,26]. This research area has also been referred to as queryable uncertainty [12] or explorable uncertainty [17]. For example, in the input to a sorting problem, we may be given for each input element, instead of its precise value, only an interval containing that point.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the Minimum problem is equivalent to the problem of determining the maximum element of each of the sets in S, e.g., by simply negating all the numbers involved. A motivation for studying the Minimum problem thus arises from the minimum spanning tree problem with uncertain edge weights [11,14,17,26]: Determining the maximum-weight edge of each cycle of a given graph allows one to determine a minimum spanning tree. Therefore, there is a connection between the problem of determining the maximum of each set in a family of possibly overlapping sets (which could be the edge sets of the cycles of a given graph) and the minimum spanning tree problem.…”

Section: Introductionmentioning

confidence: 99%

“…After this initial foundation, many classic discrete problems were studied in this framework, including geometric problems [7,9], shortest paths [15], network verification [4], minimum spanning tree [11,14,17,26], cheapest set and minimum matroid base [13,28], linear programming [25,30], traveling salesman [32], knapsack [19], and scheduling [2,3,10]. The concept of witness sets was proposed by Bruce et al [7], and identified as a pattern in many algorithms by Erlebach and Hoffmann [12].…”

Section: Introductionmentioning

confidence: 99%

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Round-Competitive Algorithms for Uncertainty Problems with Parallel Queries

Erlebach¹,

Hoffmann²,

Lima³

2021

Preprint

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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 k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. We use competitive analysis and present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds. Given a set of uncertain elements and a family of m subsets of that set, we present an algorithm for determining the value of the minimum of each of the subsets that requires at most, where opt k is the optimal number of rounds, as well as nearly matching lower bounds. For the problem of determining the i-th smallest value and identifying all elements with that value in a set of uncertain elements, we give a 2-round-competitive algorithm. We also show that the problem of sorting a family of sets of uncertain elements admits a 2-round-competitive algorithm and this is the best possible. ACM Subject ClassificationTheory of Computation → Design and analysis of algorithms; Mathematics of computing → Discrete mathematics; Theory of computation → Theory and algorithms for application domains

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“…discovery [5], shortest paths [18,41], minimum spanning tree and minimum matroid base [14,16,17,20,35,36,41], linear programming [39,34], and NP-hard problems such as the knapsack [22], scheduling [3,13] and traveling salesman problems [41]. See [15] for a survey.…”

Section: Introductionmentioning

confidence: 99%

Query Minimization under Stochastic Uncertainty

Chaplick,

Halldórsson,

de Lima

et al. 2020

Preprint

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We study problems with stochastic uncertainty information on intervals for which the precise value can be queried by paying a cost. The goal is to devise an adaptive decision tree to find a correct solution to the problem in consideration while minimizing the expected total query cost. We show that sorting in this scenario can be performed in polynomial time. For the problem of finding the data item with minimum value, we have some evidence for hardness. This contradicts intuition, since the minimum problem is easier both in the online setting with adversarial inputs and in the offline verification setting. However, the stochastic assumption can be leveraged to beat both deterministic and randomized approximation lower bounds for the online setting.

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Minimum Spanning Tree under Explorable Uncertainty in Theory and Experiments

Cited by 5 publications

References 0 publications

Query-Competitive Sorting with Uncertainty

Query-Competitive Sorting with Uncertainty

Round-Competitive Algorithms for Uncertainty Problems with Parallel Queries

Query Minimization under Stochastic Uncertainty

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