A new algorithm for encoding and decoding the Hilbert order

An iterative algorithm is described, based on the replication process of the Hilbert matrix, for encoding and decoding the Hilbert order. The time complexity of the proposed algorithm is smaller than those published previously, and the space complexity is bounded by a constant. Moreover, the new algorithm has a wider applicability when compared with existing algorithms for certain machine-word lengths. A new variant of the Hilbert curve is suggested to overcome a shortcoming of the traditional Hilbert curve for the mapping problem. The proposed coding algorithms for the traditional Hilbert curve are also applicable to the new variant without increasing the time and space complexities. Copyright c © 2006 John Wiley & Sons, Ltd.