Problem 19: Sketching vs. Streaming

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Suggested by D. Sivakumar
Source Kanpur 2006
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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:

  1. One-Way Communication: The player knowing $x$ sends a single message to the player knowing $y$ who then has to compute $f(x,y)$.
  2. 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 but cannot be computed in the simultaneous model. See [FeldmanMSSS-06] for further details.