Dynamic Online Multiselection in Internal and External Memory

Barbay, Jérémy; Gupta, Ankur; Satti, Srinivasa Rao; Sorenson, Jonathan

doi:10.1007/978-3-319-15612-5_18

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…We can further support additional insertions, deletions and queries. Data structures for online dynamic multiple selection were previously known [BGRSS15,BGSS16], but the way we handle dynamism is more efficient, allowing for all the use cases mentioned here. We discuss this in Section 2.…”

Section: Example Scenariosmentioning

confidence: 99%

“…, r q are given online and in any order have also been studied, however [BGJ + 13]. Barbay et al [BGRSS15,BGSS16] further extend this model to a dynamic setting: They consider online dynamic multiple selection where every insertion is preceded by a search for the inserted element. Deletions are ultimately performed in O(log n) time.…”

Section: Online Dynamic Multiple Selectionmentioning

confidence: 99%

“…A binary search tree and an efficient priority queue are special cases of our data structure that result when queries are frequent and uniformly distributed or only for the minimum or maximum element, respectively. While previous work has considered binary search trees in a "lazy" setting (known as "deferred data structures") [KMR88,CMS90] and multiple selection in a dynamic setting [BGRSS15,BGSS16], no existing attempts fully distinguish between insertion and query operations, severely limiting the generality of their approaches. The model we consider gives all existing results as corollaries, unifying several research directions and providing more efficient runtimes in many cases, all with the use of a single data structure.…”

Section: Introductionmentioning

confidence: 99%

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Lazy Search Trees

Sandlund,

Wild

2020

Preprint

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We introduce the lazy search tree data structure. The lazy search tree is a comparisonbased data structure on the pointer machine that supports order-based operations such as rank, select, membership, predecessor, successor, minimum, and maximum while providing dynamic operations insert, delete, change-key, split, and merge. We analyze the performance of our data structure based on a partition of current elements into a set of gaps {∆ i } based on rank. A query falls into a particular gap and splits the gap into two new gaps at a rank r associated with the query operation. If we define B = ∑ i ∆ i log 2 (n ∆ i ), our performance over a sequence of n insertions and q distinct queries is O(B + min(n log log n, n log q)). We show B is a lower bound.Effectively, we reduce the insertion time of binary search trees from Θ(log n) to O(min(log(n ∆ i ) + log log ∆ i , log q)), where ∆ i is the gap in which the inserted element falls. Over a sequence of n insertions and q queries, a time bound of O(n log q + q log n) holds; better bounds are possible when queries are non-uniformly distributed. As an extreme case of non-uniformity, if all queries are for the minimum element, the lazy search tree performs as a priority queue with O(log log n) time insert and decrease-key operations. The same data structure supports queries for any rank, interpolating between binary search trees and efficient priority queues.Lazy search trees can be implemented to operate mostly on arrays, requiring only O(min(q, n)) pointers, suggesting smaller memory footprint, better constant factors, and better cache performance compared to many existing efficient priority queues or binary search trees. Via direct reduction, our data structure also supports the efficient access theorems of the splay tree, providing a powerful data structure for non-uniform element access, both when the number of accesses is small and large.

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Section: Example Scenariosmentioning

confidence: 99%

Section: Online Dynamic Multiple Selectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lazy Search Trees

Sandlund,

Wild

2020

Preprint

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“…, l h ), but a refined analysis might. Minor improvements of our results could be brought by studying the problem in external memory, where deferred data structures have also been developed [24,2], or when the alphabet size is larger than two, as in the original article from Huffman [13].…”

Section: Perspectivesmentioning

confidence: 99%

“…. , A[j]} the set of values of an array A indexed within this range 2. We use the terminology of messages for the input and symbols for the output, as introduced by Huffman[13], which should not be confused with other terminologies found in the literature, of input symbols, letters or words for the input and output symbols or bits in the binary case.…”

mentioning

confidence: 99%

Optimal Prefix Free Codes with Partial Sorting

Barbay

2019

Algorithms

Self Cite

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We describe an algorithm computing an optimal prefix free code for n unsorted positive weights in time within O(n(1 + lg α)) ⊆ O(n lg n), where the alternation α ∈ [1..n − 1] measures the amount of sorting required by the computation. This asymptotical complexity is within a constant factor of the optimal in the algebraic decision tree computational model, in the worst case over all instances of size n and alternation α. Such results refine the state of the art complexity of Θ(n lg n) in the worst case over instances of size n in the same computational model, a landmark in compression and coding since 1952, by the mere combination of van Leeuwen's algorithm to compute optimal prefix free codes from sorted weights (known since 1976), with Deferred Data Structures to partially sort a multiset depending on the queries on it (known since 1988).

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Near-optimal online multiselection in internal and external memory

Barbay

Gupta

Satti

et al. 2016

Journal of Discrete Algorithms

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We introduce an online version of the multiselection problem, in which q selection queries are requested on an unsorted array of nelements. We provide the first online algorithm that is 1-competitive with the offline algorithm proposed by Kaligosi et al. [14] in terms of comparison complexity. Our algorithm also supports online searchqueries efficiently. We then extend our algorithm to the dynamic setting, while retaining online functionality, by supporting arbitrary insertions and deletions on the array. Assuming that the insertion of an element is immediately preceded by a search for that element, our dynamic online algorithm performs an optimal number of comparisons, up to lower order terms and an additive O(n) term. For the external memory model, we describe the first online multiselection algorithm that is O(1)-competitive. This result improves upon the work of Sibeyn [20] when q = omega(m(1-epsilon)) for any fixed positive constant epsilon, where m is the number of blocks that can be stored in main memory. We also extend it to support searches, insertions, and deletions of elements efficiently

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Dynamic Online Multiselection in Internal and External Memory

Cited by 4 publications

References 13 publications

Lazy Search Trees

Lazy Search Trees

Optimal Prefix Free Codes with Partial Sorting

Near-optimal online multiselection in internal and external memory

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