### Abstract

We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line I in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ We present two data structures for this problem. The first data structure uses O(n^{1+ε}) space and preprocessing time, and answers queries in O(2^{1/ε} log n) time. The second data structure uses O(n log^{3} n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n^{2}) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position, This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make ⊖(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

Original language | English (US) |
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Title of host publication | LATIN 2006: Theoretical Informatics - 7th Latin American Symposium, Proceedings |

Pages | 80-92 |

Number of pages | 13 |

Volume | 3887 LNCS |

DOIs | |

State | Published - 2006 |

Event | LATIN 2006: Theoretical Informatics - 7th Latin American Symposium - Valdivia, Chile Duration: Mar 20 2006 → Mar 24 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3887 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | LATIN 2006: Theoretical Informatics - 7th Latin American Symposium |
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Country | Chile |

City | Valdivia |

Period | 3/20/06 → 3/24/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*LATIN 2006: Theoretical Informatics - 7th Latin American Symposium, Proceedings*(Vol. 3887 LNCS, pp. 80-92). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3887 LNCS). https://doi.org/10.1007/11682462_12

**Data structures for halfplane proximity queries and incremental Voronoi diagrams.** / Aronov, Boris; Bose, Prosenjit; Demaine, Erik D.; Gudmundsson, Joachim; Iacono, John; Langerman, Stefan; Smid, Michiel.

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

*LATIN 2006: Theoretical Informatics - 7th Latin American Symposium, Proceedings.*vol. 3887 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3887 LNCS, pp. 80-92, LATIN 2006: Theoretical Informatics - 7th Latin American Symposium, Valdivia, Chile, 3/20/06. https://doi.org/10.1007/11682462_12

}

TY - GEN

T1 - Data structures for halfplane proximity queries and incremental Voronoi diagrams

AU - Aronov, Boris

AU - Bose, Prosenjit

AU - Demaine, Erik D.

AU - Gudmundsson, Joachim

AU - Iacono, John

AU - Langerman, Stefan

AU - Smid, Michiel

PY - 2006

Y1 - 2006

N2 - We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line I in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ We present two data structures for this problem. The first data structure uses O(n1+ε) space and preprocessing time, and answers queries in O(21/ε log n) time. The second data structure uses O(n log3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position, This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make ⊖(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

AB - We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line I in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ We present two data structures for this problem. The first data structure uses O(n1+ε) space and preprocessing time, and answers queries in O(21/ε log n) time. The second data structure uses O(n log3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position, This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make ⊖(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n) pointer changes per operation while keeping O(log n)-time point-location queries.

UR - http://www.scopus.com/inward/record.url?scp=33745587774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745587774&partnerID=8YFLogxK

U2 - 10.1007/11682462_12

DO - 10.1007/11682462_12

M3 - Conference contribution

AN - SCOPUS:33745587774

SN - 354032755X

SN - 9783540327554

VL - 3887 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 80

EP - 92

BT - LATIN 2006: Theoretical Informatics - 7th Latin American Symposium, Proceedings

ER -