### Abstract

A base-monotone region with a base is a subset of the cells in a pixel grid such that if a cell is contained in the region then so are the ones on a shortest path from the cell to the base. The problem of decomposing a pixel grid into disjoint base-monotone regions was first studied in the context of image segmentation. It is known that for a given pixel grid and base-lines, one can compute in polynomial time a maximum-weight region that can be decomposed into disjoint base-monotone regions with respect to the given base-lines (Chun et al., 2012 [4]). We continue this line of research and show the NP-hardness of the problem of optimally locating k base-lines in a given n×n pixel grid. We then present an O(n^{3})-time 2-approximation algorithm for this problem. We also study two related problems, the k base-segment problem and the quad-decomposition problem, and present some complexity results for them.

Original language | English |
---|---|

Pages (from-to) | 71-84 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 555 |

Issue number | C |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*555*(C), 71-84. https://doi.org/10.1016/j.tcs.2013.11.030

**Base-object location problems for base-monotone regions.** / Chun, Jinhee; Horiyama, Takashi; Ito, Takehiro; Kaothanthong, Natsuda; Ono, Hirotaka; Otachi, Yota; Tokuyama, Takeshi; Uehara, Ryuhei; Uno, Takeaki.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 555, no. C, pp. 71-84. https://doi.org/10.1016/j.tcs.2013.11.030

}

TY - JOUR

T1 - Base-object location problems for base-monotone regions

AU - Chun, Jinhee

AU - Horiyama, Takashi

AU - Ito, Takehiro

AU - Kaothanthong, Natsuda

AU - Ono, Hirotaka

AU - Otachi, Yota

AU - Tokuyama, Takeshi

AU - Uehara, Ryuhei

AU - Uno, Takeaki

PY - 2014/1/1

Y1 - 2014/1/1

N2 - A base-monotone region with a base is a subset of the cells in a pixel grid such that if a cell is contained in the region then so are the ones on a shortest path from the cell to the base. The problem of decomposing a pixel grid into disjoint base-monotone regions was first studied in the context of image segmentation. It is known that for a given pixel grid and base-lines, one can compute in polynomial time a maximum-weight region that can be decomposed into disjoint base-monotone regions with respect to the given base-lines (Chun et al., 2012 [4]). We continue this line of research and show the NP-hardness of the problem of optimally locating k base-lines in a given n×n pixel grid. We then present an O(n3)-time 2-approximation algorithm for this problem. We also study two related problems, the k base-segment problem and the quad-decomposition problem, and present some complexity results for them.

AB - A base-monotone region with a base is a subset of the cells in a pixel grid such that if a cell is contained in the region then so are the ones on a shortest path from the cell to the base. The problem of decomposing a pixel grid into disjoint base-monotone regions was first studied in the context of image segmentation. It is known that for a given pixel grid and base-lines, one can compute in polynomial time a maximum-weight region that can be decomposed into disjoint base-monotone regions with respect to the given base-lines (Chun et al., 2012 [4]). We continue this line of research and show the NP-hardness of the problem of optimally locating k base-lines in a given n×n pixel grid. We then present an O(n3)-time 2-approximation algorithm for this problem. We also study two related problems, the k base-segment problem and the quad-decomposition problem, and present some complexity results for them.

UR - http://www.scopus.com/inward/record.url?scp=84926279139&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84926279139&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2013.11.030

DO - 10.1016/j.tcs.2013.11.030

M3 - Article

VL - 555

SP - 71

EP - 84

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - C

ER -