Invertible Bloom Lookup Tables with Less Memory and Randomness
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In this work we study Invertible Bloom Lookup Tables (IBLTs) with small failure probabilities. IBLTs are highly versatile data structures that have found applications in set reconciliation protocols, error-correcting codes, and even the design of advanced cryptographic primitives. For storing n elements and ensuring correctness with probability at least 1 - δ, existing IBLT constructions require Ω(n(log(1/δ)/log(n)+1)) space and they crucially rely on fully random hash functions. We present new constructions of IBLTs that are simultaneously more space efficient and require less randomness. For storing n elements with a failure probability of at most δ, our data structure only requires 𝒪(n + log(1/δ)loglog(1/δ)) space and 𝒪(log(log(n)/δ))-wise independent hash functions. As a key technical ingredient we show that hashing n keys with any k-wise independent hash function h:U → [Cn] for some sufficiently large constant C guarantees with probability 1 - 2^-Ω(k) that at least n/2 keys will have a unique hash value. Proving this is highly non-trivial as k approaches n. We believe that the techniques used to prove this statement may be of independent interest.
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