O(1) Time Generation of Adjacent Multiset Combinations

We solve the problem of designing an O(1) time algorithm for generating adjacent multiset combinations in a different approach from Walsh. By the word adjacent, we mean that two adjacent multiset combinations are different at two places by one in their vector forms. Previous O(1) time algorithms for multiset combinations generated non-adjacent multiset combinations. Our algorithm in this paper can be derived from a general framework of combinatorial Gray code, which we characterise to suit our need for combinations and multiset combinations. The central idea is a twisted lexico tree, which is obtained from the lexicographic tree for the given set of combinatorial objects by twisting branches depending on the parity of each node. An iterative algorithm which traverses this tree will generate the given set of combinatorial objects in constant time as well as with a fixed number of changes from the present combinatorial object to the next

Multi-level Loop-less Algorithm for Multi-set Permutations

We present an algorithm that generates multiset permutations in O(1) time for each permutation, that is, by a loop-less algorithm with O(n) extra memory requirement. There already exist several such algorithms that generate multiset permutations in various orders. For multiset permutations, we combine two loop-less algorithms that are designed in the same principle of tree traversal. Our order of generation is different from any existing order, and the algorithm is simpler and faster than the previous ones. We also apply the new algorithm to parking functions