This paper studies variable-length (VL) source coding of general sources withside-information. Novel one-shot coding theorems for coding with commonside-information available at the encoder and the decoder and Slepian- Wolf(SW) coding (i.e., with side-information only at the decoder) are given, andthen, are applied to asymptotic analyses of these coding problems. Especially,a general formula for the infimum of the coding rate asymptotically achievableby weak VL-SW coding (i.e., VL-SW coding with vanishing error probability) isderived. Further, the general formula is applied to investigating weak VL-SWcoding of mixed sources. Our results derive and extend several known results onSW coding and weak VL coding, e.g., the optimal achievable rate of VL-SW codingfor mixture of i.i.d. sources is given for countably infinite alphabet casewith mild condition. In addition, the usefulness of the encoderside-information is investigated. Our result shows that if the encoderside-information is useless in weak VL coding then it is also useless even inthe case where the error probability may be positive asymptotically.
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