### Abstract

A code design problem for memory devices with restricted state transitions is formulated as a combinatorial optimization problem that is called a subgraph domatic partition (subDP) problem. If any neighbor set of a given state transition graph contains all the colors, then the coloring is said to be valid. The goal of a subDP problem is to find the valid coloring that has the largest number of colors for a subgraph of a given directed graph. The number of colors in an optimal valid coloring indicates the writing capacity of that state transition graph. The subDP problems are computationally hard; it is proved to be NP-complete in this paper. One of our main contributions in this paper is to show the asymptotic behavior of the writing capacity C(G) for sequences of dense bidirectional graphs; this is given by C(G) = Ω(n/ ln n), where n is the number of nodes. A probabilistic method, Lovász local lemma (LLL), plays an essential role in deriving the asymptotic expression.

Original language | English |
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Title of host publication | Proceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 1307-1311 |

Number of pages | 5 |

ISBN (Electronic) | 9781467377041 |

DOIs | |

Publication status | Published - Sep 28 2015 |

Event | IEEE International Symposium on Information Theory, ISIT 2015 - Hong Kong, Hong Kong Duration: Jun 14 2015 → Jun 19 2015 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2015-June |

ISSN (Print) | 2157-8095 |

### Other

Other | IEEE International Symposium on Information Theory, ISIT 2015 |
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Country | Hong Kong |

City | Hong Kong |

Period | 6/14/15 → 6/19/15 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modelling and Simulation
- Applied Mathematics

### Cite this

*Proceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015*(pp. 1307-1311). [7282667] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2015-June). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2015.7282667

**Subgraph domatic problem and writing capacity of memory devices with restricted state transitions.** / Wadayama, Tadashi; Izumi, Taisuke; Ono, Hirotaka.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015.*, 7282667, IEEE International Symposium on Information Theory - Proceedings, vol. 2015-June, Institute of Electrical and Electronics Engineers Inc., pp. 1307-1311, IEEE International Symposium on Information Theory, ISIT 2015, Hong Kong, Hong Kong, 6/14/15. https://doi.org/10.1109/ISIT.2015.7282667

}

TY - GEN

T1 - Subgraph domatic problem and writing capacity of memory devices with restricted state transitions

AU - Wadayama, Tadashi

AU - Izumi, Taisuke

AU - Ono, Hirotaka

PY - 2015/9/28

Y1 - 2015/9/28

N2 - A code design problem for memory devices with restricted state transitions is formulated as a combinatorial optimization problem that is called a subgraph domatic partition (subDP) problem. If any neighbor set of a given state transition graph contains all the colors, then the coloring is said to be valid. The goal of a subDP problem is to find the valid coloring that has the largest number of colors for a subgraph of a given directed graph. The number of colors in an optimal valid coloring indicates the writing capacity of that state transition graph. The subDP problems are computationally hard; it is proved to be NP-complete in this paper. One of our main contributions in this paper is to show the asymptotic behavior of the writing capacity C(G) for sequences of dense bidirectional graphs; this is given by C(G) = Ω(n/ ln n), where n is the number of nodes. A probabilistic method, Lovász local lemma (LLL), plays an essential role in deriving the asymptotic expression.

AB - A code design problem for memory devices with restricted state transitions is formulated as a combinatorial optimization problem that is called a subgraph domatic partition (subDP) problem. If any neighbor set of a given state transition graph contains all the colors, then the coloring is said to be valid. The goal of a subDP problem is to find the valid coloring that has the largest number of colors for a subgraph of a given directed graph. The number of colors in an optimal valid coloring indicates the writing capacity of that state transition graph. The subDP problems are computationally hard; it is proved to be NP-complete in this paper. One of our main contributions in this paper is to show the asymptotic behavior of the writing capacity C(G) for sequences of dense bidirectional graphs; this is given by C(G) = Ω(n/ ln n), where n is the number of nodes. A probabilistic method, Lovász local lemma (LLL), plays an essential role in deriving the asymptotic expression.

UR - http://www.scopus.com/inward/record.url?scp=84969760084&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969760084&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2015.7282667

DO - 10.1109/ISIT.2015.7282667

M3 - Conference contribution

AN - SCOPUS:84969760084

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1307

EP - 1311

BT - Proceedings - 2015 IEEE International Symposium on Information Theory, ISIT 2015

PB - Institute of Electrical and Electronics Engineers Inc.

ER -