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.