Generating Combinations by Prefix Shifts

We present a new Gray code for combinations that is practical and elegant. We represent combinations as bitstrings with s 0’s and t 1’s, and generate them with a remarkably simple rule: Identify the shortest prefix ending in 010 or 011 (or the entire bitstring if no such prefix exists) and then rotate (shift) it by one position to the right. Since the rotated portion of the string consists of at most four contiguous runs of 0’s and 1’s, each successive combination can be generated by transposing only one or two pairs of bits. This leads to a very efficient loopless implementation. The Gray code also has a simple and efficient ranking algorithm that closely resembles that of combinations in colex order. For this reason, we have given a nickname to our order: cool-lex! 1 Background and motivation An important class of computational tasks is the listing of fundamental combinatorial structures such as permutations, combinations, trees, and so on. Regarding combinations, Donald E. Knuth [8] writes “Even the apparently lowly topic of combination generation turns out to be surprisingly rich, .... I strongly believe in building up a firm foundation, so I have discussed this topic much more thoroughly than I will be able to do with material that is newer or less basic.” The applications of combination generation are numerous and varied, and Gray codes for them are particularly valuable. We mention as application areas cryptography (where they have been implemented in hardware at NSA), genetic algorithms, software and hardware Dept. of Computer Science, University of Victoria, research supported in part by NSERC. e-mail: [email protected] Dept. of Computer Science, University of Victoria, research supported in part by a NSERC PGS-D. e-mail: [email protected]