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We investigate the complexity of joint source- channel maximum a posteriori (MAP) decoding of a Markov sequence which is first encoded by a source code, then encoded by a convolutional code, and sent through a noisy memoryless channel. As established previously the MAP decoding can be performed by a Viterbi-like algorithm on a trellis whose states are triples of the states of the Markov source, source coder and convolutional coder. The large size of the product space (in the order of K2N, where K is the number of source symbols and N is the number of states of the convolutional coder) appears to prohibit such a scheme. We show that for finite impulse response convolutional codes, the state space size of joint source-channel decoding can be reduced to O(K2+N log N), hence the decoding time becomes O(TK2 +TN log N), where T is the length in bits of the decoded bitstream. We further prove that an additional complexity reduction can be achieved when K > N, if the logarithm of the source transition probabilities satisfy the so- called Monge property. This decrease becomes more significant as the tree structure of the source code is more unbalanced. The reduction factor ranges between O(K/N) (for a fixed-length source code) and O(K / log N) (for Golomb-Rice code).