Editing Open Problems:72
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 3: | Line 3: | ||
|who=Qin Zhang | |who=Qin Zhang | ||
}} | }} | ||
| − | For $\varepsilon\in(0,1]$ and $n\geq 1$, consider the following communication complexity problem $\mathrm{SIJ}_{n,\varepsilon}$: Alice and Bob are given matrices $A,B\in\{0,1\}^{n\times n}$ | + | For $\varepsilon\in(0,1]$ and $n\geq 1$, consider the following communication complexity problem $\mathrm{SIJ}_{n,\varepsilon}$: Alice and Bob are respectively given matrices $A,B\in\{0,1\}^{n\times n}$, and wish to approximate the number of non-zero elements of the product $C=AB$: i.e., to output an $(1+\varepsilon)$-approximation of $\operatorname{nnz}(C)$. |
| − | What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ ( | + | What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ (for probability of error $\delta$?) |
| − | + | What is known {{cite|GuchtWWZ-15}}: | |
| − | + | - $R^{\to}_{1/n}(\mathrm{SIJ}_{n,\varepsilon}) = \tilde{O}(\frac{n}{\varepsilon^2})$ (one-way communication) | |
| − | + | - $R_\delta(\mathrm{SIJ}_{n,\varepsilon}) = \Omega(\frac{n}{\varepsilon^{2/3}})$ for some absolute constant $\delta > 0$. | |
What is the right dependence on $\varepsilon$? | What is the right dependence on $\varepsilon$? | ||
| − | '''Note:''' $\mathrm{SIJ}$ stands for | + | '''Note:''' $\mathrm{SIJ}$ stands for "Set-Intersection Join," which is the motivation for this question. |
