Hellinger Strikes Back: A Note on the Multi-party Information Complexity of AND

The ${\textsc{And}}$ problem on t bits is a promise decision problem where either at most one bit of the input is set to 1 ( No instance) or all t bits are set to 1 (${\textsc{Yes}}$ instance). In this note, I will give a new proof of an ***(1/t ) lower bound on the information complexity of ${\textsc{And}}$ in the number-in-hand model of communication. This was recently established by Gronemeier, STACS 2009. The proof exploits the information geometry of communication protocols via Hellinger distance in a novel manner and avoids the analytic approach inherent in previous work. As previously known, this bound implies an ***(n /t ) lower bound on the communication complexity of multiparty disjointness and consequently a ***(n 1 *** 2/k ) space lower bound on estimating the k -th frequency moment F k .

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