Protocols for Asymmetric Communication Channels

Abstract

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 prove 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, the expected number of bits that must be exchanged by the server and client on the distribution D.

Efficient and Compact Representations of Prefix Codes

“…Multiplicative approximations have the potential of yielding codes that can be represented within o(n) bits. Adler and Maggs [1] showed it generally takes more than (9/40)n 1/(20c) lg n bits to store a prefix code with average codeword length at most cH(P ). Gagie [19,20,21] showed that, for any constant c ≥ 1, it takes O n 1/c log n bits to store a prefix code with average codeword length at most cH(P ) + 2.…”

“…Our specific contributions are the following. 1. In Section 3 we show that it is possible to store an optimal prefix code within O(n log max ) bits, where max = O(min(n, log N )) is the maximum length of a code (Theorem 1).…”

On the Complexity of Searching in Trees: Average-Case Minimization

“…Strategies for searching in trees have also potential application to asymmetric communication protocols [1,3,15,22,31]. In this scenario, a client has to send a binary string x ∈ {0, 1} t to the server.…”

“…In order to benefit from this discrepancy, both parties agree on a protocol to exchange bits until the server learns the string x, trying to minimize the number of bits sent by the client (though other factors, e.g., the number of rounds should also be taken into account). In one of the first protocols [3,22], at each round the server sends a binary string y and the client replies with a 0 or 1 depending on whether y is a prefix of x or not. Based on the client's answer, the server updates his knowledge about x and sends another string if he has not learned x yet.…”

Fast and Compact Prefix Codes

“…In 1998 Adler and Maggs [1] showed it generally takes more than (9/40)n 1/(20c) log n bits to store a prefix code with expected codeword length at most cH(P ), where H(P ) is P 's entropy and a lower bound on the expected codeword length. (In this paper we consider only binary codes, and by log we always mean log 2 .)…”