On O(1) time algorithms for combinatorial generation

Takaoka has recently proposed two loopless algorithms for generating well f ormed parenthesis strings with length 2n and combinations C(r, n) of n elem ents out of r elements, respectively. O(1) time algorithms for generating C (r, n) in canonical representation that were found in the literature cannot list combinations with the Strong Minimal Change Property (SMCP), Eades an d McKay's algorithm can generate C(r, n) in canonical representation with S MCP, but needs O(n) worst-case time per combination. In this paper, we main ly discuss some orders with SMCP for C(r, n) in canonical representation, a nd give a loopless algorithm to generate C(r, n) in canonical representatio n with SMCP. We also give a loopless algorithm to list well formed parenthe sis strings with length 2n. Our two algorithms are more efficient in both s pace and time than Takaoka's two algorithms, respectively. In addition, our algorithms can be modified easily to generate objects in some different or ders.