Distribution of distinguishable objects to bins: generating all distributions

Abstract

In this paper we give an algorithm to generate all distributions of distinguishable objects to bins without repetition. Our algorithm generates each distribution in constant time. To the best of our knowledge, our algorithm is the first algorithm which generates each solution in O(1) time in the ordinary sense. As a byproduct of our algorithm, we obtain a new algorithm to enumerate all multiset partitions when the number of partitions is fixed and the partitions are numbered. In this case, the algorithm generates each multiset partitions in constant time (in the ordinary sense). Finally, we extend the algorithm to the case when the bins have priorities associated with them. Overall space complexity of the algorithm is O(mklgn), where there are m bins and the objects fall into k different classes. In a companion paper, the generation of all distributions of identical objects to bins is also considered.

Keywords:

• Combinatorial objects
• Algorithm
• Generating problems
• Multiset
• Set partitions

Acknowledgements

We wish to thank the two anonymous referees for their valuable comments and suggestions for improving the presentation of the paper and for correcting the space complexity and figure 7.

Additional information

Notes on contributors

Md. Saidur Rahman

Email: [email protected]