3258 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 12, DECEMBER 2003 [6] T. M. Cover, “Broadcast channels,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 2–14, Jan. 1972. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [8] T. M. Cover, “Comments on broadcast channels,” IEEE Trans. Inform. Theory, vol. 44, pp. 2524–2530, Oct. 1998. [9] M. H. A. Davis, “Capacity and cutoff rate for the Poisson-type chan- nels,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 710–715, Nov. 1980. [10] M. R. Frey, “Information capacity of the Poisson channel,” IEEE Trans. Inform. Theory, vol. 37, pp. 244–259, Mar. 1991. [11] R. G. Gallager, “A simple derivation of the coding theorem and some applications,” IEEE Trans. Inform. Theory, vol. IT-11, pp. 3–18, Jan. 1965. [12] , “Capacity and coding for degraded broadcast channels,” Probl. Pered. Inform., vol. 10, no. 3, pp. 3–14, 1974. [13] Y. M. Kabanov, “The capacity of a channel of a Poisson type,” Theory Prob. Appl., vol. 23, pp. 143–147, 1978. [14] A. Lapidoth, “On the reliability function of the ideal Poisson channel with noiseless feedback,” IEEE Trans. Inform. Theory, vol. 39, pp. 491–503, Mar. 1993. [15] A. Lapidoth and S. Shamai (Shitz), “The Poisson multiple-access channel,” IEEE Trans. Inform. Theory, vol. 44, pp. 472–501, Mar. 1998. [16] , “How perfect need ‘Perfect side information’ be?,” IEEE Trans. Inform. Theory, vol. 48, pp. 1118–1134, May 2002. [17] A. Lapidoth, ˙ I. E. Telatar, and R. Urbanke, “On wide band broadcast channels,” in Proc. IEEE Int. Symp. Information Theory (ISIT’98), Cam- bridge, MA, Aug. 1998, p. 188. [18] J. L. Massey, “Capacity, cutoff rate and coding for direct-detection op- tical channel,” IEEE Trans. Commun., vol. COM-29, pp. 1616–1621, Nov. 1981. [19] R. J. McEliece and L. Swanson, “A note on the wide-band Gaussian broadcast channel,” IEEE Trans. Commun., vol. COM-35, pp. 452–453, Apr. 1987. [20] E. C. Posner, “Strategies for weather-dependent data acquisition,” IEEE Trans. Commun., vol. COM-31, pp. 509–517, Apr. 1983. [21] S. Shamai (Shitz), “On the capacity of a pulse amplitude modulated di- rect detection photon channel,” Proc. Inst. Elec. Eng., pt. I, vol. 137, no. 6, pp. 424–430, Dec. 1990. [22] , “On the capacity of a direct detection photon channel with inter- transition constrained binary input,” IEEE Trans. Inform. Theory, vol. 37, pp. 1540–1550, Nov. 1991. [23] S. Shamai (Shitz) and A. Lapidoth, “Bounds on the capacity of a spec- trally constrained Poisson channel,” IEEE Trans. Inform. Theory, vol. 39, pp. 19–29, Jan. 1993. [24] S. Verdú, “On channel capacity per unit cost,” IEEE Trans. Inform. Theory, vol. 36, pp. 1019–1030, Sept. 1990. [25] , “Recent results on the capacity of wideband channels in the low- power regime,” IEEE Wireless Commun., vol. 1, pp. 40–45, Aug. 2002. [26] S. Verdú, G. Caire, and D. Tuninetti, “Is TDMA optimal in the low power regime?,” in Proc. IEEE Int. Symp. Information Theory (ISIT’02), Lau- sanne, Switzerland, June/July 2002, p. 193. [27] A. D. Wyner, “Capacity and error exponent for the direct detection photon channel—Parts I and II,” IEEE Trans. Inform. Theory, vol. 34, pp. 1449–1471, Nov. 1988. On Witsenhausen’s Zero-Error Rate for Multiple Sources Gábor Simonyi Abstract—We investigate the problem of minimum rate zero-error source coding when there are several decoding terminals having different side information about the central source variable and each of them should decode in an error-free manner. For one decoder this problem was consid- ered by Witsenhausen. The Witsenhausen rate of the investigated multiple source is the asymptotically achievable minimum rate. We prove that the Witsenhausen rate of a multiple source equals the Witsenhausen rate of its weakest element. The proof relies on a powerful result of Gargano, Körner, and Vaccaro about the zero-error capacity of the compound channel. Index Terms—Chromatic number, side information, source coding, Wit- senhausen’s rate, zero error. I. INTRODUCTION Let be discrete random variables. Consider as a “central” variable available for a transmitter and the ’s as side information available for different stations that are located at different places. The joint distribution is known for and for every . The task is that broadcasts a message received by all ’s in such a way that learning this message all ’s should be able to determine in an error-free manner. The question is the minimum number of bits that should be used for this per transmission if block coding is allowed. This problem is considered for by Witsenhausen in [20]. He translated the problem to a graph-theoretic one and showed that block coding can indeed help in decreasing the (per transmission) number of possible messages that should be used. The optimal number of bits to be sent per transmission defines a graph parameter that is called Witsenhausen’s zero-error rate in [1]. (We will write simply Witsenhausen rate in the sequel.) In this correspondence, we define the Witsenhausen rate of a family of graphs. Our main result is that the Witsenhausen rate of a family of graphs equals its obvious lower bound: the largest Witsenhausen rate of the graphs in the family. This will easily follow from a powerful result of Gargano, Körner, and Vaccaro [8]. II. THE GRAPH THEORY MODEL For each we define the following graph . The vertex set is the support set of the variable for every . Two elements, and , of form an edge in if and only if there exists some possible value of the variable that is jointly pos- sible with both and , i.e., It is already explained in [20] that the minimum number of bits to be sent by to (one) for making it learn (for one instance) in an error-free manner is , where denotes the chromatic number of graph . Indeed, if would use fewer bits, than there would be some two elements of that would form an edge in and still would send the same message when one or the other would appear as the ac- tual value of . Since they form an edge, there is some possible value Manuscript received August 10, 2001; revised July 1, 2003. This work was supported in part by the Hungarian Foundation for Scientific Research under Grants (OTKA) F023442, T029255, T032323, and T037486. The material in this correspondence was presented at the IEEE International Symposium on In- formation Theory, Lausanne, Switzerland, June/July 2002. The author is with the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Hungary (e-mail: simonyi@renyi.hu). Communicated by ˙ I. E. Telatar, Associate Editor for Shannon Theory. Digital Object Identifier 10.1109/TIT.2003.820048 0018-9448/03$17.00 © 2003 IEEE