In this work, we focus on a specific practical constraint on sequential-type decoding algorithms, that is, finite stack size. Under such a practical constraint, the path deletion policy that is required when the stack exceeds its upper limit becomes essential in performance and decoding complexity. We then examined several path deletion schemes for sequential-type decoding algorithms that can produce decoding outputs in an on-the-fly fashion. Our result indicates that path deletion based on Fano metric in most cases can achieve better performance when the memory saving is critical in system design. In case the decoding process is allowed to start after the reception of the entire received word, we proposed an alternative path deletion scheme based on a two-pass decoding structure, in which the backward pass estimates the heuristic function in terms of the M-algorithm for use of the forward decoding search. As the M-algorithm can be hardware-implemented, only the computational complexity of the forward pass is accounted. Simulation results show that the computational complexity of the forward pass not only outperforms the stack algorithm with Fano metric but is smaller than that of the two-pass super-code decoder proposed in .