By developing and applying a broad framework for rejection sampling using auxiliary randomness, we provide an extension of the perfect sampling algorithm of Fill (1998) to general chains on quite general state spaces, and describe how use of bounding processes can ease computational burden. Along the way, we unearth a simple connection between the Coupling From The Past (CFTP) algorithm originated by Propp and Wilson (1996) and our extension of Fill's algorithm. Key words and phrases. Fill's algorithm, Markov chain Monte Carlo, perfect sampling, exact sampling, rejection sampling, interruptibility, coupling from the past, read-once coupling from the past, monotone transition rule, realizable monotonicity,
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