### Abstract

Consider a pair of plane straight-line graphs whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such a pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains, one of which is convex, in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^{3}n) time and O(n + m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n disjoint axis-aligned rectangles in O(n log^{2}n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.

Original language | English (US) |
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Title of host publication | Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings |

Publisher | Springer Verlag |

Pages | 473-484 |

Number of pages | 12 |

Volume | 10389 LNCS |

ISBN (Print) | 9783319621265 |

DOIs | |

State | Published - 2017 |

Event | 15th International Symposium on Algorithms and Data Structures, WADS 2017 - St. John’s, Canada Duration: Jul 31 2017 → Aug 2 2017 |

### 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 | 10389 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th International Symposium on Algorithms and Data Structures, WADS 2017 |
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Country | Canada |

City | St. John’s |

Period | 7/31/17 → 8/2/17 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings*(Vol. 10389 LNCS, pp. 473-484). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10389 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62127-2_40

**Searching edges in the overlap of two plane graphs.** / Iacono, John; Khramtcova, Elena; Langerman, Stefan.

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

*Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings.*vol. 10389 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10389 LNCS, Springer Verlag, pp. 473-484, 15th International Symposium on Algorithms and Data Structures, WADS 2017, St. John’s, Canada, 7/31/17. https://doi.org/10.1007/978-3-319-62127-2_40

}

TY - GEN

T1 - Searching edges in the overlap of two plane graphs

AU - Iacono, John

AU - Khramtcova, Elena

AU - Langerman, Stefan

PY - 2017

Y1 - 2017

N2 - Consider a pair of plane straight-line graphs whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such a pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains, one of which is convex, in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log3n) time and O(n + m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n disjoint axis-aligned rectangles in O(n log2n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.

AB - Consider a pair of plane straight-line graphs whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such a pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains, one of which is convex, in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log3n) time and O(n + m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n disjoint axis-aligned rectangles in O(n log2n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.

UR - http://www.scopus.com/inward/record.url?scp=85025118036&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85025118036&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-62127-2_40

DO - 10.1007/978-3-319-62127-2_40

M3 - Conference contribution

AN - SCOPUS:85025118036

SN - 9783319621265

VL - 10389 LNCS

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

SP - 473

EP - 484

BT - Algorithms and Data Structures - 15th International Symposium, WADS 2017, Proceedings

PB - Springer Verlag

ER -