Difference between revisions of "Open Problems:10"
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Consider the communication problem '''Gap-Hamdist''': Alice and Bob are given length $n$ binary strings $x$ and $y$ such that either the Hamming distance $\Delta(x,y) \leq n/2$ or $\Delta(x,y)\geq n/2+\sqrt{n}$. The one-way communication complexity of '''Gap-Hamdist''' is known to be $\Omega(n)$ {{cite|IndykW-03|Woodruff-04}}. Recently, a simpler proof was discovered using a reduction from '''Index''' {{cite|JayramKS-07}}. Is the multi-round communication complexity also $\Omega(n)$? There is a $\Omega(\sqrt{n})$ lower-bound from a reduction from '''Set-Disjointness''' but we conjecture that the lower-bound is actually $\Omega(n)$. | Consider the communication problem '''Gap-Hamdist''': Alice and Bob are given length $n$ binary strings $x$ and $y$ such that either the Hamming distance $\Delta(x,y) \leq n/2$ or $\Delta(x,y)\geq n/2+\sqrt{n}$. The one-way communication complexity of '''Gap-Hamdist''' is known to be $\Omega(n)$ {{cite|IndykW-03|Woodruff-04}}. Recently, a simpler proof was discovered using a reduction from '''Index''' {{cite|JayramKS-07}}. Is the multi-round communication complexity also $\Omega(n)$? There is a $\Omega(\sqrt{n})$ lower-bound from a reduction from '''Set-Disjointness''' but we conjecture that the lower-bound is actually $\Omega(n)$. | ||
If the conjecture is true then it would imply stronger multiple-pass lower bounds for estimating $F_0$ {{cite|IndykW-03|Woodruff-04|BarYossefJKST-02}} and entropy {{cite|BhuvanagiriG-06|ChakrabartiCM-07}}. Alternatively, if the conjecture is not true then it would be interesting to see if better multi-pass algorithms exist for $F_0$ and entropy. | If the conjecture is true then it would imply stronger multiple-pass lower bounds for estimating $F_0$ {{cite|IndykW-03|Woodruff-04|BarYossefJKST-02}} and entropy {{cite|BhuvanagiriG-06|ChakrabartiCM-07}}. Alternatively, if the conjecture is not true then it would be interesting to see if better multi-pass algorithms exist for $F_0$ and entropy. | ||
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| + | ==Update== | ||
| + | This conjecture was proved by Chakrabarti and Regev {{cite|ChakrabartiR-11}}, who showed that the communication complexity of '''Gap-Hamdist''' is $\Omega(n)$. | ||
Latest revision as of 04:03, 28 April 2017
| Suggested by | Ravi Kumar |
|---|---|
| Source | Kanpur 2006 |
| Short link | https://sublinear.info/10 |
Consider the communication problem Gap-Hamdist: Alice and Bob are given length $n$ binary strings $x$ and $y$ such that either the Hamming distance $\Delta(x,y) \leq n/2$ or $\Delta(x,y)\geq n/2+\sqrt{n}$. The one-way communication complexity of Gap-Hamdist is known to be $\Omega(n)$ [IndykW-03,Woodruff-04]. Recently, a simpler proof was discovered using a reduction from Index [JayramKS-07]. Is the multi-round communication complexity also $\Omega(n)$? There is a $\Omega(\sqrt{n})$ lower-bound from a reduction from Set-Disjointness but we conjecture that the lower-bound is actually $\Omega(n)$.
If the conjecture is true then it would imply stronger multiple-pass lower bounds for estimating $F_0$ [IndykW-03,Woodruff-04,BarYossefJKST-02] and entropy [BhuvanagiriG-06,ChakrabartiCM-07]. Alternatively, if the conjecture is not true then it would be interesting to see if better multi-pass algorithms exist for $F_0$ and entropy.
Update[edit]
This conjecture was proved by Chakrabarti and Regev [ChakrabartiR-11], who showed that the communication complexity of Gap-Hamdist is $\Omega(n)$.
