Nearly Optimal Static Las Vegas Succinct Dictionary

For positive integers $U$, $n$, and $\sigma$, given a set $S$ of $n$ (distinct) keys from key space $[U]$, each associated with value $[\sigma]$, the static dictionary problem asks one to preprocess these (key, value) pairs into data structure, supporting value-retrieval queries: for any $x\in [U]$, ${valRet}(x)$ must return $x$ if S$, or $\bot$ $x\notin S$. The special case where $\sigma=1$ is called membership problem. “textbook” solution use hash table, which occupies linear answers query in constant time. On other hand, minimum possible encode all only ${OPT}:= \lceil\lg_2\binom{U}{n}+n\lg_2 \sigma\rceil$ bits, could be much smaller than table. In this paper, we design randomized structure using ${OPT}+{poly}\lg n+O(\lg^{(\ell)} U)$ bits space, it has expected time, assuming algorithm can access an external lookup table size $n^{\epsilon}$ $\ell$ $\epsilon$. depends on not input. Previously, even queries $U\leq n^{O(1)}$, best known time requires ${OPT}+n/{poly}\lg n$ (Pagh [SIAM J. Comput., 31 (2001), pp. 353--363] P?tra?cu [FOCS, IEEE Computer Society, Los Alamitos, CA, 2008, 305--313]); ${OPT}+n^{1-\epsilon}$ $O(\lg n)$. Our new open questions by Thorup 305--313; Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 109 (2013), 7--13]. We also present scheme that compresses sequence $X\in[\sigma]^n$ its zeroth order (empirical) entropy up $\sigma\cdot{poly}\lg extra decoding $X_i$ \sigma)$