A method for measuring dissimilarities between cyclic strings is introduced. It computes a weighted mean between two (lower and upper) bounds of the exact cyclic edit distance, which are founded on a window-constrained edit graph related to the strings involved. Weights are the ones which minimize the sum of squared relative errors of the weighted solution with respect to exact values, on a training set of string pairs. This method takes ???????? time. Experiments on both artificial and real data, show the highly accurate solutions achieved by this technique, which is clearly faster than the most efficient exact algorithms.