Bloom Filters, Adaptivity, and the Dictionary Problem

TL;DR

This paper investigates the limitations and possibilities of adaptive approximate membership query data structures, demonstrating the impossibility of small adaptive AMQs and proposing an optimal, adaptive solution with efficient operations.

Contribution

It proves the impossibility of small adaptive AMQs and introduces a new adaptive AMQ design that is space-efficient and supports constant-time queries and updates.

Findings

Impossibility of small adaptive AMQs proved

Proposed adaptive AMQ with optimal space usage

Supports worst-case constant-time queries and updates

Abstract

The Bloom filter---or, more generally, an approximate membership query data structure (AMQ)---maintains a compact, probabilistic representation of a set S of keys from a universe U. An AMQ supports lookups, inserts, and (for some AMQs) deletes. A query for an x in S is guaranteed to return "present." A query for x not in S returns "absent" with probability at least 1-epsilon, where epsilon is a tunable false positive probability. If a query returns "present," but x is not in S, then x is a false positive of the AMQ. Because AMQs have a nonzero probability of false-positives, they require far less space than explicit set representations. AMQs are widely used to speed up dictionaries that are stored remotely (e.g., on disk/across a network). Most AMQs offer weak guarantees on the number of false positives they will return on a sequence of queries. The false-positive probability of epsilon holds only for a single query. It is easy for an adversary to drive an AMQ's false-positive rate towards 1 by simply repeating false positives. This paper shows what it takes to get strong guarantees on the number of false positives. We say that an AMQs is adaptive if it guarantees a false-positive probability of epsilon for every query, regardless of answers to previous queries. First, we prove that it is impossible to build a small adaptive AMQ, even when the AMQ is immediately told whenever it returns a false positive. We then show how to build an adaptive AMQ that partitions its state into a small local component and a larger remote component. In addition to being adaptive, the local component of our AMQ dominates existing AMQs in all regards. It uses optimal space up to lower-order terms and supports queries and updates in worst-case constant time, with high probability. Thus, we show that adaptivity has no cost.