Abstract-We consider a lossless multi-terminal source coding problem with one transmitter, two receivers and side information. The achievable rate region of the problem is not well understood. In this paper, we characterise the rate region when the side information at one receiver is conditionally less noisy than the side information at the other, given this other receiver's desired source. The conditionally less noisy definition includes degraded side information and a common message as special cases, and it is motivated by the concept of less noisy broadcast channels. The key contribution of the paper is a new converse theorem employing a telescoping identity and the Csiszár sum identity.
I. INTRODUCTION AND PROBLEM STATEMENTConsider the multi-terminal source coding problem shown in Fig. 1. A discrete memoryless source emits an independent and identically distributed (iid) sequence of correlated random variables (X, Y, U, V ). The Transmitter observes the (X, Y )-component, Receiver 1 observes the U -component, and Receiver 2 observes the V -component. The Transmitter jointly compresses X and Y to a binary stream of rate R, and it sends this stream over a noiseless channel to both receivers. We wish to determine the smallest rate, R * , at which Receivers 1 and 2 can reliably recover the X and Y -components respectively.The described problem is a special case of the rate-distortion functions in . In this paper, we determine R * for the case where H(Y |U ) ≤ H(Y |V ) and the side information U at Receiver 1 is conditionally less noisy than the side information V at Receiver 2 given Y . Our definition of conditionally less noisy side information includes (i) and (iii) as special cases. The definition is motivated by the less noisy condition for discrete memoryless broadcast channels [4], [5]. The key contribution of the paper is a new converse theorem for this class of sources. The converse makes use of a telescoping identity [6] and the Csiszár sum identity [5, Sec. 2.3].We now describe the problem statement more formally. Let X , Y, U and V denote the finite alphabets of X, Y , U and V respectively. We identify the n-fold Cartesian product of these alphabets using boldfaced notation; for example, X is the n-fold product of X .