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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