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Recent advances in DNA sequencing technology and DNA storage systems haverekindled the interest in deletion channels. Multiple recent works have lookedat variants of sequence reconstruction over a single and over multiple deletionchannels, a notoriously difficult problem due to its highly combinatorialnature. Although works in theoretical computer science have provided algorithmswhich guarantee perfect reconstruction with multiple independent observationsfrom the deletion channel, they are only applicable in the large blocklengthregime and more restrictively, when the number of observations is also large.Indeed, with only a few observations, perfect reconstruction of the inputsequence may not even be possible in most cases. In such situations, maximumlikelihood (ML) and maximum aposteriori (MAP) estimates for the deletionchannels are natural questions that arise and these have remained open to thebest of our knowledge. In this work, we take steps to answer the twoaforementioned questions. Specifically: 1. We show that solving for the MLestimate over the single deletion channel (which can be cast as a discreteoptimization problem) is equivalent to solving its relaxation, a continuousoptimization problem; 2. We exactly compute the symbolwise posteriordistributions (under some assumptions on the priors) for both the single aswell as multiple deletion channels. As part of our contributions, we alsointroduce tools to visualize and analyze error events, which we believe couldbe useful in other related problems concerning deletion channels.

Algorithms for reconstruction over single and multiple deletion channels