Engineering Predecessor Data Structures for Dynamic Integer Sets

TL;DR

This paper develops highly optimized data structures for the dynamic predecessor problem, improving speed and space efficiency for maintaining integer sets under updates and queries, with practical implementations for common bit widths.

Contribution

The authors engineer and implement optimized variants of universe reduction, y-fast tries, and B-trees, demonstrating superior performance and space efficiency over previous methods.

Findings

Data structures outperform previous approaches in speed and space.

Implementations for 32, 40, 64-bit integers are practical and efficient.

Insights on data storage strategies and CPU instruction benefits.

Abstract

We present highly optimized data structures for the dynamic predecessor problem, where the task is to maintain a set $S$ of $w$-bit numbers under insertions, deletions, and predecessor queries (return the largest element in $S$ no larger than a given key). The problem of finding predecessors can be viewed as a generalized form of the membership problem, or as a simple version of the nearest neighbour problem. It lies at the core of various real-world problems such as internet routing. In this work, we engineer (1) a simple implementation of the idea of universe reduction, similar to van-Emde-Boas trees (2) variants of y-fast tries [Willard, IPL'83], and (3) B-trees with different strategies for organizing the keys contained in the nodes, including an implementation of dynamic fusion nodes [P\v{a}tra\c{s}cu and Thorup, FOCS'14]. We implement our data structures for $w=32,40,64$, which covers most typical scenarios. Our data structures finish workloads faster than previous approaches while being significantly more space-efficient, e.g., they clearly outperform standard implementations of the STL by finishing up to four times as fast using less than a third of the memory. Our tests also provide more general insights on data structure design, such as how small sets should be stored and handled and if and when new CPU instructions such as advanced vector extensions pay off.