We reformulate an interpolation-based unique decoding algorithm of AG codes, using the theory of Gr$ddot{o}$bner bases of modules on the coordinate ring of the base curve. The conceptual description of the reformulated algorithm lets us better understand the majority voting procedure, which is central in the interpolation-based unique decoding. Moreover the smaller Gr$ddot{o}$bner bases imply smaller space and time complexity of the algorithm.