The cylinder arrays are main load-carrying structures in offshore engineering. Most analyses on the hydrodynamics of cylinder arrays usually consider the instance that cylinder arrays react to loads as one whole body, namely that there is no relative movements between each cylinder of the arrays. Analysis on arrays of cylinders, among which relative motions occur, is comparatively rare. Therefore, the diffraction-radiation problem of arrays of bottom-seated vertical circular cylinders with relative motions among cylinders is analyzed by an exact algebraic method in this paper. Firstly, the radiation problem of cylinder arrays among which each cylinder moves with prscribed independent amplitude is studied. Then the diffraction-radiation problem of cylinder arrays with relative movement in the presence of incident waves is analyzed. In addition, the dydrodynamic forces on the cylinders and oscillation amplitudes of each cylinder are obtained. As a validation, the degenerate case is calculated and compared with the pulished one. And further calculations indicate that the influences of the relative movements among cylinders on the hydrodynamic responses are obvious in some range of wave numbers.