Improved bounds for asymmetric communication protocols
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In this paper we examine the problem of sending an $n$-bit data item from a client to a server across an asymmetric communication channel. We demonstrate that there are scenarios in which a high-speed link from the server to the client can be used to greatly reduce the number of bits sent from the client to the server across a slower link. In particular, we assume that the data item is drawn from a probability distribution $D$ that is known to the server but not to the client. We present several protocols in which the expected number of bits transmitted by the server and client are $O(n)$ and $O(H(D)+1)$, respectively, where $H(D)$ is the binary entropy of $D$ (and can range from $0$ to $n$). These protocols are within a small constant factor of optimal in terms of the number of bits sent by the client. The expected number of rounds of communication between the server and client in the simplest of our protocols is $O(H(D))$. We also give a protocol for which the expected number of rounds is only $O(1)$, but which requires more computational effort on the part of the server. A third technique provides a tradeoff between the computational effort and the number of rounds. These protocols are complemented by several lower bounds and impossibility results. We demonstrate that all of our protocols are existentially optimal in terms of the number of bits sent by the server, i.e., there are distributions for which the total number of bits exchanged has to be at least $n$. In addition, we show that there is no protocol that is optimal for every distribution (as opposed to just existentially optimal) in terms of bits sent by the server. We demonstrate this by proving that it is undecidable to compute (even approximately), for an arbitrary distribution $D$, how many expected bits must be exchanged by the server and client on the distribution $D$.