# A quadratically tight partition bound for classical communication complexity and query complexity

@article{Jain2014AQT, title={A quadratically tight partition bound for classical communication complexity and query complexity}, author={R. Jain and T. Lee and N. Vishnoi}, journal={ArXiv}, year={2014}, volume={abs/1401.4512} }

In this work we introduce, both for classical communication complexity and query complexity, a modification of the 'partition bound' introduced by Jain and Klauck [2010]. We call it the 'public-coin partition bound'. We show that (the logarithm to the base two of) its communication complexity and query complexity versions form, for all relations, a quadratically tight lower bound on the public-coin randomized communication complexity and randomized query complexity respectively.

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The Partition Bound for Classical Communication Complexity and Query Complexity

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A Communication protocol to realize a partition Let {R 1 , R 2 , . . . , R m } be a partition of X × Y. Let x and y be the inputs to Alice and Bob respectively

A quadratically tight partition bound ...B QUERY PROTOCOL TO REALIZE A PARTITION

A s } be a partition of {0, 1} n such that |A i | ≤ m for each i ∈ [s]. Let x be the string in the database

Alice queries the bits of x corresponding to A 1 . If the bits revealed are consistent with A 1 then she considers A 1 as desired assignment and stops

B Query protocol to realize a partition

We note that the number of such rounds is at most m and in each round at most m bits are revealed. Hence the total number of queries is at most m 2