### Abstract

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in (Formula presented.). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is (Formula presented.). A matching (Formula presented.) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the (Formula presented.) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of (Formula presented.) from the diagram of S, in time (Formula presented.) worst case, which is (Formula presented.) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in (Formula presented.) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Original language | English (US) |
---|---|

Pages (from-to) | 1-27 |

Number of pages | 27 |

Journal | Discrete and Computational Geometry |

DOIs | |

State | Accepted/In press - Oct 4 2017 |

### Fingerprint

### Keywords

- Grappa tree
- Incremental
- Link-cut
- Voronoi diagrams

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Discrete and Computational Geometry*, 1-27. https://doi.org/10.1007/s00454-017-9943-2

**Incremental Voronoi Diagrams.** / Allen, Sarah R.; Barba, Luis; Iacono, John; Langerman, Stefan.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, pp. 1-27. https://doi.org/10.1007/s00454-017-9943-2

}

TY - JOUR

T1 - Incremental Voronoi Diagrams

AU - Allen, Sarah R.

AU - Barba, Luis

AU - Iacono, John

AU - Langerman, Stefan

PY - 2017/10/4

Y1 - 2017/10/4

N2 - We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in (Formula presented.). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is (Formula presented.). A matching (Formula presented.) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the (Formula presented.) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of (Formula presented.) from the diagram of S, in time (Formula presented.) worst case, which is (Formula presented.) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in (Formula presented.) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

AB - We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in (Formula presented.). We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is (Formula presented.). A matching (Formula presented.) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the (Formula presented.) upper bound of Aronov et al. (LATIN 2006: Theoretical Informatics. Lecture Notes in Computer Science, Springer, Berlin, 2006) for farthest-point Voronoi diagrams in the special case where the points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of (Formula presented.) from the diagram of S, in time (Formula presented.) worst case, which is (Formula presented.) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in (Formula presented.) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

KW - Grappa tree

KW - Incremental

KW - Link-cut

KW - Voronoi diagrams

UR - http://www.scopus.com/inward/record.url?scp=85030670685&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030670685&partnerID=8YFLogxK

U2 - 10.1007/s00454-017-9943-2

DO - 10.1007/s00454-017-9943-2

M3 - Article

SP - 1

EP - 27

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -