Nearly Optimal Static Las Vegas Succinct Dictionary
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Given a set S of n (distinct) keys from key space [U], each associated with a value from Σ, the static dictionary problem asks to preprocess these (key, value) pairs into a data structure, supporting value-retrieval queries: for any given x∈ [U], valRet(x) must return the value associated with x if x∈ S, or return if x∉ S. The special case where |Σ|=1 is called the membership problem. The "textbook" solution is to use a hash table, which occupies linear space and answers each query in constant time. On the other hand, the minimum possible space to encode all (key, value) pairs is only OPT:= _2Un+n_2|Σ| bits, which could be much less. In this paper, we design a randomized dictionary data structure using OPT+poly n+O( U) bits of space, and it has expected constant query time, assuming the query algorithm can access an external lookup table of size n^0.001. The lookup table depends only on U, n and |Σ|, and not the input. Previously, even for membership queries and U≤ n^O(1), the best known data structure with constant query time requires OPT+n/poly n bits of space (Pagh [Pag01] and Pǎtraşcu [Pat08]); the best-known using OPT+n^0.999 space has query time O( n); the only known non-trivial data structure with OPT+n^0.001 space has O( n) query time and requires a lookup table of size ≥ n^2.99 (!). Our new data structure answers open questions by Pǎtraşcu and Thorup [Pat08,Tho13].
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