Improved Bounds for Poset Sorting in the Forbidden-Comparison Regime

Biswas, Arindam; Jayapaul, Varunkumar; Raman, Venkatesh

doi:10.1007/978-3-319-53007-9_5

Cited by 5 publications

(3 citation statements)

References 4 publications

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“…More recently, a more general form of sorting on partially ordered sets (or posets), where some pairs of elements are incomparable, has been studied [2,5,6,9,10]. Given a input set P of n elements, poset sorting algorithms determine the underlying partial order of the elements.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Sorting in Trees

Roychoudhury¹,

Yadav²

2022

Preprint

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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 than w in trees. For example, in complete binary trees, d = Θ(1), w = Θ(n). We present a randomized algorithm for sorting a tree poset in worst-case expected O(dn log n) query and time complexity. This improves the previous upper bound of O(dn log 2 n). Our algorithm is the first to be optimal for bounded-degree trees. We also provide a new lower bound of Ω(dn + n log n) for the worstcase query complexity of sorting a tree poset. Finally, we present the first deterministic algorithm for sorting tree posets that has lower total complexity than existing algorithms for sorting general partial orders.

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Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Sorting in Trees

Roychoudhury¹,

Yadav²

2022

Preprint

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“…Although there have been several subsequent papers on the topic [4,5,14] (which we will discuss further shortly), these upper bounds of Õ(n 1.5 ) and Õ(n 1.4 ) have remained the state of the art for the worst-case and stochastic generalized sorting problems, respectively. Moreover, no nontrivial lower bounds are known.…”

Section: Introductionmentioning

confidence: 99%

“…The difficulty of these problems has led researchers to consider alternative formulations that are more tractable. Banerjee and Richards [4] considered the worst-case generalized sorting problem in the setting where G is very dense, containing n 2 − q edges for some parameter q, and gave an algorithm that performs O((q + n) lg n) comparisons; whether this is optimal remains open, and the best known lower bound is Ω(q + n lg n) [5]. Banerjee and Richards [4] also gave on alternative algorithm for the stochastic generalized sorting problem, achieving Õ(min{n 3/2 , pn 2 }) comparisons, but this bound is never better than that of [11] for any p. Work by Lu et al [14] considered a variation on the worst-case generalized sorting problem in which we are also given predicted outcomes for all possible comparisons, and all but w of the predictions are guaranteed to be true.…”

Section: Introductionmentioning

confidence: 99%

Stochastic and Worst-Case Generalized Sorting Revisited

Kuszmaul¹,

Narayanan²

2021

Preprint

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The generalized sorting problem is a restricted version of standard comparison sorting where we wish to sort n elements but only a subset of pairs are allowed to be compared. Formally, there is some known graph G = (V, E) on the n elements v 1 , . . . , v n , and the goal is to determine the true order of the elements using as few comparisons as possible, where all comparisons (v i , v j ) must be edges in E. We are promised that if the true ordering is x 1 < x 2 < • • • < x n for {x i } an unknown permutation of the vertices {v i }, then (x i , x i+1 ) ∈ E for all i: this Hamiltonian path ensures that sorting is actually possible.In this work, we improve the bounds for generalized sorting on both random graphs and worstcase graphs. For Erdős-Renyi random graphs G(n, p) (with the promised Hamiltonian path added to ensure sorting is possible), we provide an algorithm for generalized sorting with an expected O(n lg(np)) comparisons, which we prove to be optimal for query complexity. This strongly improves over the best known algorithm of Huang, Kannan, and Khanna (FOCS 2011), which uses Õ(min(n √ np, n/p 2 )) comparisons. For arbitrary graphs G with n vertices and m edges (again with the promised Hamiltonian path), we provide an algorithm for generalized sorting with Õ( √ mn) comparisons. This improves over the best known algorithm of Huang et al., which uses min(m, Õ(n 3/2 )) comparisons.

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