# Problem 75: Data Structure Lower Bound in the Cell Probe Model

Suggested by | Kasper Green Larsen |
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Source | Banff 2017 |

Short link | https://sublinear.info/75 |

Input is an $n \times n$ boolean matrix $M$. We can preprocess $M$ and store a data structure. Then on query $v$, an $n$ bit vector, we need to output $Mv$, which is matrix multiplication with $\cdot$ replaced by $\wedge$ and $+$ replaced by $\vee$. The preprocessing time is denoted by $t_p$ and query time is denoted by $t_q$.

It is conjectured that in the word-RAM model, $t_p + nt_q \ge n^{3-o(1)}$. But in the cell-probe model, Larsen and Williams [LarsenW-17] give a data structure that uses space $n^2 + n^{7/4}\sqrt{w}$, i.e., just $n^{7/4}\sqrt{w}$ extra bits, where $w$ is the word size (which is typically $\Theta(\log n)$). The data structure computes $Mv$ using $t_q = n^{7/4}/\sqrt{w}$ probes in the worst case. Such a data structure that stores only a small amount of extra bits is called a *succinct* data structure.

**Question:** Can we show a lower bound of $\omega(n)$ on $t_q$ in the cell-probe model for succinct data structures?