Computer Science/Discrete Mathematics Seminar I

A dictionary data structure maintains a set of at most $n$ keys from the universe $[U]$ under key insertions and deletions, such that given a query $x\in[U]$, it returns if $x$ is in the set. Some variants also store values associated to the keys such that given a query $x$, the value associated to $x$ is returned when $x$ is in the set.

This fundamental data structure problem has been studied for six decades since the introduction of hash tables in 1953. A hash table occupies $O(n\log U)$ bits of space with constant time per operation in expectation. There has been a vast literature on improving its time and space usage. The state-of-the-art dictionary by Bender, Farach-Colton, Kuszmaul, Kuszmaul and Liu has space consumption close to the information-theoretic optimum, using a total of \[\log\binom{U}{n}+O(n\log^{(k)} n)\]
bits, while supporting all operations in $O(k)$ time, for any parameter $k\leq \log^* n$. The term $O(\log^{(k)} n)=O(\underbrace{\log\cdots\log}_k n)$ is referred to as the wasted bits per key.

In this talk, we show a matching cell-probe lower bound: For $U=n^{1+\Theta(1)}$, any dictionary with $O(\log^{(k)} n)$ wasted bits per key must have expected operational time $\Omega(k)$, in the cell-probe model with word-size $w=\Theta(\log U)$. Furthermore, if a dictionary stores values of $\Theta(\log U)$ bits, we show that regardless of the query time, it must have $\Omega(k)$ expected update time. It is worth noting that this is the first cell-probe lower bound on the trade-off between space and update time for general data structures.

Joint work with Tianxiao Li, Jingxun Liang and Renfei Zhou.