Support Optimality and Adaptive Cuckoo Filters

Abstract

Filters (such as Bloom Filters) are data structures that speed up network routing and measurement operations by storing a compressed representation of a set. Filters are space efficient, but can make bounded one-sided errors: with tunable probability epsilon, they may report that a query element is stored in the filter when it is not. This is called a false positive. Recent research has focused on designing methods for dynamically adapting filters to false positives, reducing the number of false positives when some elements are queried repeatedly. Ideally, an adaptive filter would incur a false positive with bounded probability epsilon for each new query element, and would incur o(epsilon) total false positives over all repeated queries to that element. We call such a filter support optimal. In this paper we design a new Adaptive Cuckoo Filter and show that it is support optimal (up to additive logarithmic terms) over any n queries when storing a set of size n. Our filter is simple: fixing previous false positives requires a simple cuckoo operation, and the filter does not need to store any additional metadata. This data structure is the first practical data structure that is support optimal, and the first filter that does not require additional space to fix false positives. We complement these bounds with experiments showing that our data structure is effective at fixing false positives on network traces, outperforming previous Adaptive Cuckoo Filters. Finally, we investigate adversarial adaptivity, a stronger notion of adaptivity in which an adaptive adversary repeatedly queries the filter, using the result of previous queries to drive the false positive rate as high as possible. We prove a lower bound showing that a broad family of filters, including all known Adaptive Cuckoo Filters, can be forced by such an adversary to incur a large number of false positives.