A Selectable Sloppy Heap

TL;DR

This paper introduces a selectable sloppy heap, a novel data structure that supports dynamic insertion and deletion of elements in constant time, improving efficiency in percentile and order statistic computations.

Contribution

The paper presents the first data structure that handles both insertions and deletions in constant time for fixed quantile groups, outperforming previous soft heap implementations.

Findings

Supports dynamic insertions and deletions in constant time

Outperforms soft heap in dynamic percentile maintenance

Efficient for fixed quantile group operations

Abstract

We study the selection problem, namely that of computing the $i$th order statistic of $n$ given elements. Here we offer a data structure called \emph{selectable sloppy heap} handling 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 (ideal!) 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 $0<\varepsilon \leq 1/2$) in applications such as dynamic percentile maintenance. The design demonstrates how slowing down a certain computation can speed up the data structure.