Difference between revisions of "Open Problems:75"
(Created page with "{{Header |source=banff17 |who=Kasper Green Larsen }} Input is an $n \times n$ boolean matrix $M$. We can preprocess $M$ and store a data structure. Then on query $v$, an $n$...") |
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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 $t_q$. | 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 Willimas {{cite| | + | 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 Willimas {{cite|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. |
Question is, can we show a lower bound of $\omega(n)$ on $t_q$ in the cell-probe model? | Question is, can we show a lower bound of $\omega(n)$ on $t_q$ in the cell-probe model? | ||
Revision as of 22:31, 1 April 2017
| Suggested by | Kasper Green Larsen |
|---|---|
| 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 Willimas [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.
Question is, can we show a lower bound of $\omega(n)$ on $t_q$ in the cell-probe model?
