Theory of Computing ------------------- Title : Lower Bounds for Data Structures with Space Close to Maximum Imply Circuit Lower Bounds Authors : Emanuele Viola Volume : 15 Number : 18 Pages : 1-9 URL : https://theoryofcomputing.org/articles/v015a018 Abstract -------- Let $f:{0,1}^n\to {0,1}^m$ be a function computable by a circuit with unbounded fan-in, arbitrary gates, $w$ wires and depth $d$. With a very simple argument we show that the $m$-query problem corresponding to $f$ has data structures with space $s=n+r$ and time $(w/r)^{d}$, for any $r$. As a consequence, in the setting where $s$ is close to $m$ a slight improvement on the state of existing data-structure lower bounds would solve long-standing problems in circuit complexity. We also use this connection to obtain a data structure for error- correcting codes which nearly matches the 2007 lower bound by Gal and Miltersen. This data structure can also be made dynamic. Finally we give a problem that requires at least $3$ bit probes for $m=n^{O(1)}$ and even $s=m/2-1$. Independent work by Dvir, Golovnev, and Weinstein (2018) and by Corrigan-Gibbs and Kogan (2018) give incomparable connections between data-structure and other types of lower bounds.