Sorting by pile shuffles on queue-like and stack-like piles can be hard
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909639639040000 |
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| author | Treleaven, Kyle B. |
| author_facet | Treleaven, Kyle B. |
| contents | Inspired by a common technique for shuffling a deck of cards on a table without riffling, we continue the study of a prequel paper on the pile shuffle and its capabilities as a sorting device. We study two sort feasibility problems of general interest concerning pile shuffle, first introduced in the prequel. These problems are characterized by: (1) bounds on the number of sequential rounds of shuffle, and piles created in each round; (2) the use of a heterogeneous mixture of queue-like and stack-like piles, as when each round of shuffle may have a combination of face-up and face-down piles; and (3) the ability of the dealer to choose the types of piles used during each round of shuffle. We prove by a sequence of reductions from the Boolean satisfiability problem (SAT) that the more general problem is NP-Hard. We leave as an open question the complexity of its arguably more natural companion, but discuss avenues for further investigation. Our analysis leverages a novel framework, introduced herein, which equates instances of shuffle to members of a particular class of deterministic finite automata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sorting by pile shuffles on queue-like and stack-like piles can be hard Treleaven, Kyle B. Computational Complexity Inspired by a common technique for shuffling a deck of cards on a table without riffling, we continue the study of a prequel paper on the pile shuffle and its capabilities as a sorting device. We study two sort feasibility problems of general interest concerning pile shuffle, first introduced in the prequel. These problems are characterized by: (1) bounds on the number of sequential rounds of shuffle, and piles created in each round; (2) the use of a heterogeneous mixture of queue-like and stack-like piles, as when each round of shuffle may have a combination of face-up and face-down piles; and (3) the ability of the dealer to choose the types of piles used during each round of shuffle. We prove by a sequence of reductions from the Boolean satisfiability problem (SAT) that the more general problem is NP-Hard. We leave as an open question the complexity of its arguably more natural companion, but discuss avenues for further investigation. Our analysis leverages a novel framework, introduced herein, which equates instances of shuffle to members of a particular class of deterministic finite automata. |
| title | Sorting by pile shuffles on queue-like and stack-like piles can be hard |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2506.05518 |