Near-optimal online multiselection in internal and external memory

Abstract

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

Lazy Search Trees

“…Their data structure uses B + o(B) + O(n + q ′ log n) comparisons, where q ′ is the number of search, insert, and delete operations. The crucial difference between our solution and that of Barbay et al [BGRSS15,BGSS16] is how we handle insertions. Their analysis assumes every insertion is preceded by a search and therefore insertion must take Ω(log n) time.…”

“…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.…”

“…, 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.…”

“…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.…”

Selectable Heaps and Optimal Lazy Search Trees

“…In some sense, this research approaches a unification of different ordered data structures into a single theory of a best data structure for ordered data. Optimal priority queues (a single gap ∆ 1 ), online dynamic multiple selection [BGSS16] (gaps are separated by queried ranks), and binary search trees (every element is in its own gap) are special cases of our solution. This paper closes the book theoretically in the gap-based model proposed in [SW20]; any further work in this research direction requires generalization of the gap model, stated in [SW20] as open problem 1.…”