Generating Gray Codes in O(1) Worst-Case Time per Word

We give a definition of Gray code that, unlike the standard “minimal change” definition, is satisfied by the word-lists in the literature called “Gray codes” and we give several examples to illustrate the various concepts of minimality. We show that a non-recursive generation algorithm can be obtained for a word-list such that all the words with the same prefix (or, equivalently, suffix) are consecutive and that the Bitner-Ehrlich-Reingold method of generating each word in a time bounded by a constant works under the additional condition that in the interval of words with the same prefix or suffix the next letter assumes at least two values. Finally we generalize this method so that it works under a weaker condition satisfied by almost all the Gray codes in the literature: if the next letter assumes only one value, then the interval contains only one word. 1 Definition of a Gray Code Gray codes have nothing to do with anything so sinister as the color gray, despite the fact that the first article on the subject [1] was published by the military. They were named after the author of that article, Frank Gray, who ordered the set of length-n binary strings (words on a two-letter alphabet, usually {0,1}, which represent subsets of {1, 2, ..., n}) into a list in which two consecutive strings differ in only one position (see Table 1 below), and probably also after the title: Pulse code communication. Subsequently, other sets of words that represent various combinatorial objects such as combinations or k -subsets (see Table 2 below), permutations (see Table 3 below), integer compositions, set partitions, integer partitions and binary trees (represented by Dyck words see Table 4 below) were ordered into lists in which two consecutive words differ in only a small number of positions, and these lists were called Gray codes because they share the same “minimal change” property as Gray’s pulse code. A Gray code is sometimes defined as a word-list in which the Hamming distance between any two adjacent words in the list (the number of positions in which these two words differ) is minimal. Most of the word-lists in the literature that are called Gray codes do not satisfy that definition or are published without a minimality proof. For example, if a k -subset of {1, 2, ..., n} is represented by its members in increasing order, then the Hamming distance between adjacent C.S. Calude et al. (Eds.): DMTCS 2003, LNCS 2731, pp. 73–88, 2003. c © Springer-Verlag Berlin Heidelberg 2003