A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps

Y. I Jen Chiang, Franco P. Preparata, Roberto Tamassia

    Research output: Contribution to journalArticle

    Abstract

    We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with n vertices and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in M. The space requirement is O (n log n). Point-location queries take time O (log n). Ray-shooting and shortest-path queries take time O (log3n) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3n) time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time.

    Original languageEnglish (US)
    Pages (from-to)207-233
    Number of pages27
    JournalSIAM Journal on Computing
    Volume25
    Issue number1
    StatePublished - Feb 1996

    Fingerprint

    Ray Shooting
    Planar Maps
    Point Location
    Shortest path
    Data structures
    Query
    Dynamic Data Structures
    Update
    Decomposition
    Support Point
    Deletion
    Insertion
    Data Structures
    Decompose
    Requirements
    Vertex of a graph

    Keywords

    • Computational geometry
    • Dynamic algorithm
    • Point location
    • Ray shooting
    • Shortest path

    ASJC Scopus subject areas

    • Computational Theory and Mathematics
    • Applied Mathematics
    • Theoretical Computer Science

    Cite this

    Chiang, Y. I. J., Preparata, F. P., & Tamassia, R. (1996). A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps. SIAM Journal on Computing, 25(1), 207-233.

    A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps. / Chiang, Y. I Jen; Preparata, Franco P.; Tamassia, Roberto.

    In: SIAM Journal on Computing, Vol. 25, No. 1, 02.1996, p. 207-233.

    Research output: Contribution to journalArticle

    Chiang, YIJ, Preparata, FP & Tamassia, R 1996, 'A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps', SIAM Journal on Computing, vol. 25, no. 1, pp. 207-233.
    Chiang, Y. I Jen ; Preparata, Franco P. ; Tamassia, Roberto. / A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps. In: SIAM Journal on Computing. 1996 ; Vol. 25, No. 1. pp. 207-233.
    @article{0fe164f2946e430f9694981dabcb7ec6,
    title = "A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps",
    abstract = "We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with n vertices and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in M. The space requirement is O (n log n). Point-location queries take time O (log n). Ray-shooting and shortest-path queries take time O (log3n) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3n) time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time.",
    keywords = "Computational geometry, Dynamic algorithm, Point location, Ray shooting, Shortest path",
    author = "Chiang, {Y. I Jen} and Preparata, {Franco P.} and Roberto Tamassia",
    year = "1996",
    month = "2",
    language = "English (US)",
    volume = "25",
    pages = "207--233",
    journal = "SIAM Journal on Computing",
    issn = "0097-5397",
    publisher = "Society for Industrial and Applied Mathematics Publications",
    number = "1",

    }

    TY - JOUR

    T1 - A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps

    AU - Chiang, Y. I Jen

    AU - Preparata, Franco P.

    AU - Tamassia, Roberto

    PY - 1996/2

    Y1 - 1996/2

    N2 - We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with n vertices and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in M. The space requirement is O (n log n). Point-location queries take time O (log n). Ray-shooting and shortest-path queries take time O (log3n) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3n) time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time.

    AB - We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map M with n vertices and apply it to the development of a unified dynamic data structure that supports point-location, ray-shooting, and shortest-path queries in M. The space requirement is O (n log n). Point-location queries take time O (log n). Ray-shooting and shortest-path queries take time O (log3n) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log3n) time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time.

    KW - Computational geometry

    KW - Dynamic algorithm

    KW - Point location

    KW - Ray shooting

    KW - Shortest path

    UR - http://www.scopus.com/inward/record.url?scp=0029771983&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0029771983&partnerID=8YFLogxK

    M3 - Article

    VL - 25

    SP - 207

    EP - 233

    JO - SIAM Journal on Computing

    JF - SIAM Journal on Computing

    SN - 0097-5397

    IS - 1

    ER -