Searching in Dynamic Tree-Like Partial Orders

Heeringa, Brent; Iordan, Marius Cătălin; Theran, Louis

doi:10.1007/978-3-642-22300-6_43

Cited by 5 publications

(4 citation statements)

References 16 publications

(22 reference statements)

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“…is is sometimes cast as searching in a partial order. A dynamic variant, supporting insertion and deletion of elements, has been proposed by Heeringa et al [HIT11]. Another model that was recently proposed for searching a graph involves an oracle that given a vertex, returns the first edge incident to this vertex on a shortest path to the sought vertex [EKS16].…”

Section: Related Workmentioning

confidence: 99%

Competitive Online Search Trees on Trees

Bose

Cardinal

Iacono

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree.is model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. e model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O(log log n)-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.

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Section: Related Workmentioning

confidence: 99%

Competitive Online Search Trees on Trees

Bose

Cardinal

Iacono

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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show abstract

“…Apart from the theoretical significance of the problem, there are many practical applications, for example, constructing the evolutionary tree of coronavirus strains or general phylogenetic trees, network mapping, among others. In addition, the final sorted tree can be used as input for tree searching algorithms, such as [11,15]. Onak and Parys [15] considered extending the concept of binary search to trees, whereas Heeringa, Iordan, and Lewis [11] considered searching in dynamic trees.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the final sorted tree can be used as input for tree searching algorithms, such as [11,15]. Onak and Parys [15] considered extending the concept of binary search to trees, whereas Heeringa, Iordan, and Lewis [11] considered searching in dynamic trees.…”

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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“…Related work. Different models of searching in trees have also been considered, e.g., the one where we query edges instead of vertices [BFN99, LN01, MOW08, OP06], with connections to searching in posets [LS85,HIT11]. In the edge-query setting, Cicalese, Jacobs, Laber and Molinaro [CJLM11,CJLM14] study the problem of minimizing the average search time of a vertex, and show this to be an NP-hard problem [CJLM11].…”

Section: Introductionmentioning

confidence: 99%

Fast approximation of search trees on trees with centroid trees

Berendsohn¹,

Golinsky²,

Kaplan³

et al. 2022

Preprint

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Search trees on trees (STTs) generalize the fundamental binary search tree (BST) data structure: in STTs the underlying search space is an arbitrary tree, whereas in BSTs it is a path. An optimal BST of size n can be computed for a given distribution of queries in O(n 2 ) time [Knuth, Acta Inf. 1971] and centroid BSTs provide a nearly-optimal alternative, computable in O(n) time [Mehlhorn, SICOMP 1977].By contrast, optimal STTs are not known to be computable in polynomial time, and the fastest constant-approximation algorithm runs in O(n 3 ) time [Berendsohn, Kozma, SODA 2022]. Centroid trees can be defined for STTs analogously to BSTs, and they have been used in a wide range of algorithmic applications. In the unweighted case (i.e., for a uniform distribution of queries), the centroid tree can be computed in O(n) time [Brodal, Fagerberg, Pedersen, Östlin, ICALP 2001; Della Giustina, Prezza, Venturini, SPIRE 2019]. These algorithms, however, do not readily extend to the weighted case. Moreover, no approximation guarantees were previously known for centroid trees in either the unweighted or weighted cases.In this paper we revisit centroid trees in a general, weighted setting, and we settle both the algorithmic complexity of constructing them, and the quality of their approximation. For constructing a weighted centroid tree, we give an output-sensitive O(n log h) ⊆ O(n log n) time algorithm, where h is the height of the resulting centroid tree. If the weights are of polynomial complexity, the running time is O(n log log n). We show these bounds to be optimal, in a general decision tree model of computation. For approximation, we prove that the cost of a centroid tree is at most twice the optimum, and this guarantee is best possible, both in the weighted and unweighted cases. We also give tight, fine-grained bounds on the approximation-ratio for bounded-degree trees and on the approximation-ratio of more general α-centroid trees.

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Searching in Dynamic Tree-Like Partial Orders

Cited by 5 publications

References 16 publications

Competitive Online Search Trees on Trees

Competitive Online Search Trees on Trees

Efficient Algorithms for Sorting in Trees

Fast approximation of search trees on trees with centroid trees

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