Editing Open Problems:19

{{DISPLAYTITLE:Problem 19: Sketching vs. Streaming}}
{{Infobox
|label1 = Proposed by
|data1 = D. Sivakumar
|label2 = Source
|data2 = [[Workshops:Kanpur_2006|Kanpur 2006]]
|label3 = Short link
|data3 = http://sublinear.info/19
}}
Show that any symmetric function that admits a good streaming algorithm also admits a sketching algorithm. In terms of communication complexity, consider trying to evaluate a symmetric function $f(x,y)$ in each of the following models:
#''One-Way Communication:'' The player knowing $x$ sends a single message to the player knowing $y$ who then has to compute $f(x,y)$.
#''Simultaneous Communication:'' Both players send a message simultaneously to a third party who then has to compute $f(x,y)$.

Obviously any function that can be evaluated with $B$ bits of communication in the simultaneous model can be evaluated in the one-way model with $B$ bits of communication. Are there natural functions that require significantly more communication in the simultaneous model than in the one-way model? It is known that any total, permutation-invariant function that can be computed in the one-way model can be computed in the simultaneous model. See {{cite|FeldmanMSSS-06}} for further details.