Difference between revisions of "Open Problems:72"
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| − | For $\varepsilon\in(0,1]$ and $n\geq 1$, consider the following communication complexity problem $\mathrm{SIJ}_{n,\varepsilon}$: Alice and Bob are | + | 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 $A,B\in\{0,1\}^{n\times n}$, respectively, and wish to output a $(1+\varepsilon)$-approximation to the number of non-zero entries in the product $C=AB$. |
| − | What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ ( | + | What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ (where $\delta$ is the probability of error)? |
| − | + | Known facts {{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. |
Revision as of 21:50, 12 January 2016
| Suggested by | Qin Zhang |
|---|---|
| Source | Baltimore 2016 |
| Short link | https://sublinear.info/72 |
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 $A,B\in\{0,1\}^{n\times n}$, respectively, and wish to output a $(1+\varepsilon)$-approximation to the number of non-zero entries in the product $C=AB$. What is the two-way randomized communication complexity $R_\delta(\mathrm{SIJ}_{n,\varepsilon})$ (where $\delta$ is the probability of error)?
Known facts [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 $\varepsilon$?
Note: $\mathrm{SIJ}$ stands for “Set-Intersection Join,” which is the motivation for this question.
