A Hash Table Without Hash Functions, and How to Get the Most Out of Your Random Bits
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This paper considers the basic question of how strong of a probabilistic guarantee can a hash table, storing n (1 + Θ(1)) log n-bit key/value pairs, offer? Past work on this question has been bottlenecked by limitations of the known families of hash functions: The only hash tables to achieve failure probabilities less than 1 / 2^ n require access to fully-random hash functions – if the same hash tables are implemented using the known explicit families of hash functions, their failure probabilities become 1 / (n). To get around these obstacles, we show how to construct a randomized data structure that has the same guarantees as a hash table, but that avoids the direct use of hash functions. Building on this, we are able to construct a hash table using O(n) random bits that achieves failure probability 1 / n^n^1 - ϵ for an arbitrary positive constant ϵ. In fact, we show that this guarantee can even be achieved by a succinct dictionary, that is, by a dictionary that uses space within a 1 + o(1) factor of the information-theoretic optimum. Finally we also construct a succinct hash table whose probabilistic guarantees fall on a different extreme, offering a failure probability of 1 / (n) while using only Õ(log n) random bits. This latter result matches (up to low-order terms) a guarantee previously achieved by Dietzfelbinger et al., but with increased space efficiency and with several surprising technical components.
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