Star unfolding of a polytope with applications

Pankaj K. Agarwal, Boris Aronov, Joseph O'Rourke, Catherine A. Schevon

    Research output: Contribution to journalArticle

    Abstract

    We introduce the notion of a star unfolding of the surface P of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n6β(n) log n), where β(n) is an extremely slowly growing function. A much simpler O(n6) time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an O(n8 log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter 1 ≤ m ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, into a data structure of size O(n6m1+δ), so that a query can be answered in time O((√n/m1/4) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n5m1+δ) preprocessing time and storage and guarantee O((n/m)1/3 log n) query time for any choice of m between 1 and n.

    Original languageEnglish (US)
    Pages (from-to)1689-1713
    Number of pages25
    JournalSIAM Journal on Computing
    Volume26
    Issue number6
    StatePublished - Dec 1997

    Fingerprint

    Unfolding
    Polytope
    Stars
    Star
    Shortest path
    Query
    Data structures
    Data Structures
    Convex Polytope
    Geodesic
    Preprocessing
    Three-dimensional
    Computing

    Keywords

    • Convex polytopes
    • Geodesics
    • Shortest paths
    • Star unfolding

    ASJC Scopus subject areas

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

    Cite this

    Agarwal, P. K., Aronov, B., O'Rourke, J., & Schevon, C. A. (1997). Star unfolding of a polytope with applications. SIAM Journal on Computing, 26(6), 1689-1713.

    Star unfolding of a polytope with applications. / Agarwal, Pankaj K.; Aronov, Boris; O'Rourke, Joseph; Schevon, Catherine A.

    In: SIAM Journal on Computing, Vol. 26, No. 6, 12.1997, p. 1689-1713.

    Research output: Contribution to journalArticle

    Agarwal, PK, Aronov, B, O'Rourke, J & Schevon, CA 1997, 'Star unfolding of a polytope with applications', SIAM Journal on Computing, vol. 26, no. 6, pp. 1689-1713.
    Agarwal PK, Aronov B, O'Rourke J, Schevon CA. Star unfolding of a polytope with applications. SIAM Journal on Computing. 1997 Dec;26(6):1689-1713.
    Agarwal, Pankaj K. ; Aronov, Boris ; O'Rourke, Joseph ; Schevon, Catherine A. / Star unfolding of a polytope with applications. In: SIAM Journal on Computing. 1997 ; Vol. 26, No. 6. pp. 1689-1713.
    @article{850515f858204c41ad3e39f6d3324406,
    title = "Star unfolding of a polytope with applications",
    abstract = "We introduce the notion of a star unfolding of the surface P of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n6β(n) log n), where β(n) is an extremely slowly growing function. A much simpler O(n6) time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an O(n8 log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the following form: {"}Given two points, what is the length of the shortest path connecting them?{"} Given a parameter 1 ≤ m ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, into a data structure of size O(n6m1+δ), so that a query can be answered in time O((√n/m1/4) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n5m1+δ) preprocessing time and storage and guarantee O((n/m)1/3 log n) query time for any choice of m between 1 and n.",
    keywords = "Convex polytopes, Geodesics, Shortest paths, Star unfolding",
    author = "Agarwal, {Pankaj K.} and Boris Aronov and Joseph O'Rourke and Schevon, {Catherine A.}",
    year = "1997",
    month = "12",
    language = "English (US)",
    volume = "26",
    pages = "1689--1713",
    journal = "SIAM Journal on Computing",
    issn = "0097-5397",
    publisher = "Society for Industrial and Applied Mathematics Publications",
    number = "6",

    }

    TY - JOUR

    T1 - Star unfolding of a polytope with applications

    AU - Agarwal, Pankaj K.

    AU - Aronov, Boris

    AU - O'Rourke, Joseph

    AU - Schevon, Catherine A.

    PY - 1997/12

    Y1 - 1997/12

    N2 - We introduce the notion of a star unfolding of the surface P of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n6β(n) log n), where β(n) is an extremely slowly growing function. A much simpler O(n6) time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an O(n8 log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter 1 ≤ m ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, into a data structure of size O(n6m1+δ), so that a query can be answered in time O((√n/m1/4) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n5m1+δ) preprocessing time and storage and guarantee O((n/m)1/3 log n) query time for any choice of m between 1 and n.

    AB - We introduce the notion of a star unfolding of the surface P of a three-dimensional convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n6β(n) log n), where β(n) is an extremely slowly growing function. A much simpler O(n6) time algorithm that finds a small superset of all such edge sequences is also sketched. The second algorithm is an O(n8 log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the following form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter 1 ≤ m ≤ n2, it can preprocess P in time O(n6m1+δ), for any δ > O, into a data structure of size O(n6m1+δ), so that a query can be answered in time O((√n/m1/4) log n). If one query point always lies on an edge of P, the algorithm can be improved to use O(n5m1+δ) preprocessing time and storage and guarantee O((n/m)1/3 log n) query time for any choice of m between 1 and n.

    KW - Convex polytopes

    KW - Geodesics

    KW - Shortest paths

    KW - Star unfolding

    UR - http://www.scopus.com/inward/record.url?scp=0013013983&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0013013983&partnerID=8YFLogxK

    M3 - Article

    VL - 26

    SP - 1689

    EP - 1713

    JO - SIAM Journal on Computing

    JF - SIAM Journal on Computing

    SN - 0097-5397

    IS - 6

    ER -