A Selectable Sloppy Heap

Abstract

We study the selection problem, namely that of computing the ith order statistic of n given elements. Here we offer a data structure called selectable sloppy heap that handles a dynamic version in which upon request (i) a new element is inserted or (ii) an element of a prescribed quantile group is deleted from the data structure. Each operation is executed in constant time—and is thus independent of n (the number of elements stored in the data structure)—provided that the number of quantile groups is fixed. This is the first result of this kind accommodating both insertion and deletion in constant time. As such, our data structure outperforms the soft heap data structure of Chazelle (which only offers constant amortized complexity for a fixed error rate 0 < ε ≤ 1 / 2 ) in applications such as dynamic percentile maintenance. The design demonstrates how slowing down a certain computation can speed up the data structure. The method described here is likely to have further impact in the field of data structure design in extending asymptotic amortized upper bounds to same formula asymptotic worst-case bounds.

Keywords

• dynamic quantile maintenance
• approximate selection
• ith order statistic
• mediocre element
• tournament
• online algorithm