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New Direct Product results in Communication Complexity. Rahul Jain Centre for Quantum Technologies and Department of Computer Science National University of Singapore. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A A A A A A A A. - PowerPoint PPT Presentation
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New Direct Product results in Communication Complexity
Rahul Jain
Centre for Quantum Technologies andDepartment of Computer ScienceNational University of Singapore.
Direct Product
x1; : : :;xk y1; : : : ;yk
f µ X £ Y £ Z
Let there be ERROORR!
Direct Product : Is the trivial protocol essentially optimal ?
Can we solve k copies faster ?
Open: Rpub1¡ 2¡ ( k ) (f k) = (k ¢Rpub
² (f )) ?
Task: Output z1; : : :zk 2 Z k such that for all i;(xi ;yi ;zi ) 2 f
Our Results
x1; : : :;xk y1; : : : ;yk
x1; :: : ;xk y1; :: : ;yk
f µ X £ Y £ Z
Application: Direct product for
(First shown by [Klauck 10])
One-Way Case
New Characterization
¸ has error at most ²¸ is one-way for ¹
x1x2x3...
y1 y2 y3 ¢¢¢
...
¢¢¢
rec1;¹² (f ) = minf log 1
¹ (R ) g
x1x2x3...
y1 y2 y3 ¢¢¢
...
¢¢¢
New Characterization
Use message compression of [Braverman, Rao 10] to get a one-way protocol with communication
Message Compression[Braverman, Rao 10]
Direct Product
Two-way casef µ X £ Y £ Z
¸ has error at most ²
x1x2x3...
y1 y2 y3
rec¹² (f ) = minf log 1
¹ (R ) g
¢¢¢
...
¢¢¢x1x2x3...
y1 y2 y3 ¢¢¢
...
¢¢¢
¸ is message-like for ¹
ApplicationDirect Product for Set Disjointness
[Razborov 92]
Our result
Thanks !
Questions ?Holy Grail: Rpub
1¡ 2¡ ( k ) (f k) = (k ¢Rpub² (f )) ?
En route:
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