A Necessary and Sufficient Condition for Reconstruction of Circular Binary String based on the Lengths of Substrings with Weights

A circular string, which is made by connecting the first and the last symbol of a string, is one of simple sequence forms and is used for many applications such as data compression and fragment assembly problem. A sufficient condition on the lengths of substrings with weights for reconstruction of an input circular binary string is shown. However, there are no detailed descriptions on the proof of the sufficient condition and reconstruction algorithm.In this paper, we prove a necessary and sufficient condition on the lengths of substrings with weights for reconstruction of the circular string. We show the length is shorter than that of previous study for some circular strings. For improving the length, we use minimal absent words (MAWs) for given substrings of length k, and we propose a new construction algorithm of MAWs of length h (> k) while a conventional construction algorithm of MAWs can construct MAWs of length ℓ (≤ k). Moreover, we propose reconstruction algorithm of an input circular string for given substrings satisfying the new condition.