Incremental Voronoi diagrams

Sarah R. Allen, Luis Barba, John Iacono, Stefan Langerman

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (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 ℝ3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(√n). A matching Ω(√n) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log n) upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams in the special case where 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 S ∪ {p} from the diagram of S, in time O(K polylog n) worst case, which is O(√n polylog n) 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 O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

    Original languageEnglish (US)
    Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
    PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
    Pages15.1-15.16
    Volume51
    ISBN (Electronic)9783959770095
    DOIs
    StatePublished - Jun 1 2016
    Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
    Duration: Jun 14 2016Jun 17 2016

    Other

    Other32nd International Symposium on Computational Geometry, SoCG 2016
    CountryUnited States
    CityBoston
    Period6/14/166/17/16

    Fingerprint

    Data structures
    Triangulation

    Keywords

    • Delaunay triangulation
    • Dynamic data structures
    • Voronoi diagrams

    ASJC Scopus subject areas

    • Software

    Cite this

    Allen, S. R., Barba, L., Iacono, J., & Langerman, S. (2016). Incremental Voronoi diagrams. In 32nd International Symposium on Computational Geometry, SoCG 2016 (Vol. 51, pp. 15.1-15.16). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2016.15

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

    32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. p. 15.1-15.16.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Allen, SR, Barba, L, Iacono, J & Langerman, S 2016, Incremental Voronoi diagrams. in 32nd International Symposium on Computational Geometry, SoCG 2016. vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 15.1-15.16, 32nd International Symposium on Computational Geometry, SoCG 2016, Boston, United States, 6/14/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.15
    Allen SR, Barba L, Iacono J, Langerman S. Incremental Voronoi diagrams. In 32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2016. p. 15.1-15.16 https://doi.org/10.4230/LIPIcs.SoCG.2016.15
    Allen, Sarah R. ; Barba, Luis ; Iacono, John ; Langerman, Stefan. / Incremental Voronoi diagrams. 32nd International Symposium on Computational Geometry, SoCG 2016. Vol. 51 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. pp. 15.1-15.16
    @inproceedings{fc6c7d48fa0e41bbbe550c7f3b6ca775,
    title = "Incremental Voronoi diagrams",
    abstract = "We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (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 ℝ3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(√n). A matching Ω(√n) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log n) upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams in the special case where 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 S ∪ {p} from the diagram of S, in time O(K polylog n) worst case, which is O(√n polylog n) 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 O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.",
    keywords = "Delaunay triangulation, Dynamic data structures, Voronoi diagrams",
    author = "Allen, {Sarah R.} and Luis Barba and John Iacono and Stefan Langerman",
    year = "2016",
    month = "6",
    day = "1",
    doi = "10.4230/LIPIcs.SoCG.2016.15",
    language = "English (US)",
    volume = "51",
    pages = "15.1--15.16",
    booktitle = "32nd International Symposium on Computational Geometry, SoCG 2016",
    publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

    }

    TY - GEN

    T1 - Incremental Voronoi diagrams

    AU - Allen, Sarah R.

    AU - Barba, Luis

    AU - Iacono, John

    AU - Langerman, Stefan

    PY - 2016/6/1

    Y1 - 2016/6/1

    N2 - We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (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 ℝ3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(√n). A matching Ω(√n) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log n) upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams in the special case where 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 S ∪ {p} from the diagram of S, in time O(K polylog n) worst case, which is O(√n polylog n) 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 O(log n) 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 VD(S) (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 ℝ3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(√n). A matching Ω(√n) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log n) upper bound of Aronov et al. (2006) for farthest-point Voronoi diagrams in the special case where 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 S ∪ {p} from the diagram of S, in time O(K polylog n) worst case, which is O(√n polylog n) 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 O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

    KW - Delaunay triangulation

    KW - Dynamic data structures

    KW - Voronoi diagrams

    UR - http://www.scopus.com/inward/record.url?scp=84976891131&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84976891131&partnerID=8YFLogxK

    U2 - 10.4230/LIPIcs.SoCG.2016.15

    DO - 10.4230/LIPIcs.SoCG.2016.15

    M3 - Conference contribution

    VL - 51

    SP - 15.1-15.16

    BT - 32nd International Symposium on Computational Geometry, SoCG 2016

    PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

    ER -