### Uittreksel

The Euclidean k-Center problem is a classical problem that has been extensively studied in computer science. Given a set G of n points in Euclidean space, the problem is to determine a set C of k centers (not necessarily part of G) such that the maximum distance between a point in G and its nearest neighbor in C is minimized. In this paper we study the corresponding (k, ℓ)-CENTER problem for polygonal curves under the Fréchet distance, that is, given a set G of n polygonal curves in ℝd, each of complexity m, determine a set C of k polygonal curves in ℝd, each of complexity ℓ, such that the maximum Fréchet distance of a curve in G to its closest curve in C is minimized. In their 2016 paper, Driemel, Krivošija, and Sohler give a near-linear time (1 + ε-approximation algorithm for one-dimensional curves, assuming that k and ℓ are constants. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. Our analysis thus extends to application-relevant input data such as GPS-trajectories and protein backbones. We show that, if ℓ is part of the input, then there is no polynomial-time approximation scheme unless P = NP. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k = 1, and the NP-hardness extends to the case that ℓ = ∞, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez’ algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.181

Taal | Engels |
---|---|

Titel | 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) |

Redacteuren | Timothy M. Chan |

Uitgeverij | Society for Industrial and Applied Mathematics (SIAM) |

Pagina's | 2922-2938 |

Aantal pagina's | 17 |

ISBN van elektronische versie | 978-1-61197-548-2 |

DOI's | |

Status | Gepubliceerd - 2 jan 2019 |

Evenement | ACM-SIAM Symposium on Discrete Algorithms (SODA2019) - San Diego, Verenigde Staten van Amerika Duur: 6 jan 2019 → 9 jan 2019 https://www.siam.org/Conferences/CM/Conference/soda19 |

### Congres

Congres | ACM-SIAM Symposium on Discrete Algorithms (SODA2019) |
---|---|

Verkorte titel | SODA2019 |

Land | Verenigde Staten van Amerika |

Stad | San Diego |

Periode | 6/01/19 → 9/01/19 |

Internet adres |

### Vingerafdruk

### Citeer dit

*30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)*(blz. 2922-2938). Society for Industrial and Applied Mathematics (SIAM). DOI: 10.1137/1.9781611975482.181

}

*30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).*Society for Industrial and Applied Mathematics (SIAM), blz. 2922-2938, San Diego, Verenigde Staten van Amerika, 6/01/19. DOI: 10.1137/1.9781611975482.181

**Approximating (k,ℓ)-center clustering for curves.** / Buchin, Kevin; Driemel, Anne; Gudmundsson, Joachim; Horton, Michael; Kostitsyna, Irina; Löffler, Maarten; Struijs, Martijn.

Onderzoeksoutput: Hoofdstuk in Boek/Rapport/Congresprocedure › Hoofdstuk › Academic › peer review

TY - CHAP

T1 - Approximating (k,ℓ)-center clustering for curves

AU - Buchin,Kevin

AU - Driemel,Anne

AU - Gudmundsson,Joachim

AU - Horton,Michael

AU - Kostitsyna,Irina

AU - Löffler,Maarten

AU - Struijs,Martijn

PY - 2019/1/2

Y1 - 2019/1/2

N2 - The Euclidean k-Center problem is a classical problem that has been extensively studied in computer science. Given a set G of n points in Euclidean space, the problem is to determine a set C of k centers (not necessarily part of G) such that the maximum distance between a point in G and its nearest neighbor in C is minimized. In this paper we study the corresponding (k, ℓ)-CENTER problem for polygonal curves under the Fréchet distance, that is, given a set G of n polygonal curves in ℝd, each of complexity m, determine a set C of k polygonal curves in ℝd, each of complexity ℓ, such that the maximum Fréchet distance of a curve in G to its closest curve in C is minimized. In their 2016 paper, Driemel, Krivošija, and Sohler give a near-linear time (1 + ε-approximation algorithm for one-dimensional curves, assuming that k and ℓ are constants. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. Our analysis thus extends to application-relevant input data such as GPS-trajectories and protein backbones. We show that, if ℓ is part of the input, then there is no polynomial-time approximation scheme unless P = NP. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k = 1, and the NP-hardness extends to the case that ℓ = ∞, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez’ algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.181

AB - The Euclidean k-Center problem is a classical problem that has been extensively studied in computer science. Given a set G of n points in Euclidean space, the problem is to determine a set C of k centers (not necessarily part of G) such that the maximum distance between a point in G and its nearest neighbor in C is minimized. In this paper we study the corresponding (k, ℓ)-CENTER problem for polygonal curves under the Fréchet distance, that is, given a set G of n polygonal curves in ℝd, each of complexity m, determine a set C of k polygonal curves in ℝd, each of complexity ℓ, such that the maximum Fréchet distance of a curve in G to its closest curve in C is minimized. In their 2016 paper, Driemel, Krivošija, and Sohler give a near-linear time (1 + ε-approximation algorithm for one-dimensional curves, assuming that k and ℓ are constants. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. Our analysis thus extends to application-relevant input data such as GPS-trajectories and protein backbones. We show that, if ℓ is part of the input, then there is no polynomial-time approximation scheme unless P = NP. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k = 1, and the NP-hardness extends to the case that ℓ = ∞, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez’ algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.181

U2 - 10.1137/1.9781611975482.181

DO - 10.1137/1.9781611975482.181

M3 - Chapter

SP - 2922

EP - 2938

BT - 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

PB - Society for Industrial and Applied Mathematics (SIAM)

ER -