Bloom Filters, Adaptivity, and the Dictionary Problem
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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 lookup, and possibly insert and delete operations. If x in S, then lookup(x) returns "present." If x not in S, then, lookup(x) may return "present" with probability at most epsilon, where epsilon is a tunable false-positive probability, and such an x is called a false positive of the AMQ. Otherwise lookup(x) returns "absent." AMQs have become widely used to accelerate dictionaries that are stored remotely (e.g., on disk or across a network). By using an AMQ, the dictionary needs to access the remote representation of S only when the AMQ indicates that the queried item might be present in S. Thus, the primary goal of an AMQ is to minimize its false-positive rate, so that the number of unnecessary accesses to the remote representation of S can be minimized. However, the false-positive guarantees for AMQs are rather weak. The false-positive probability of epsilon holds only for distinct or randomly chosen queries, but does not hold for arbitrary sequences of queries. For example, an adversary that chooses its queries based on the outcomes of previous queries can easily create a sequence of queries consisting almost entirely of false positives. Even simply repeating a randomly chosen query has an epsilon chance of producing a sequence entirely of false positives. In this paper, we give adaptive AMQs that do have strong false-positive guarantees. In particular, for any fixed epsilon, our AMQs guarantee a false-positive rate of epsilon for every query and for every sequence of previously made queries. Furthermore, our adaptive AMQ is optimal in terms of space (up to lower order terms) and complexity (all operations are constant time).
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