Exact size of binary space partitionings and improved rectangle tiling algorithms

Piotr Berman, Bhaskar Dasgupta, S. Muthukrishnan

    Research output: Contribution to journalArticle

    15 Citations (Scopus)

    Abstract

    We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n - 1 in general, and an upper bound of 2n - 1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Windmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255-259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. • A worst-case lower bound of 2n - o(n) in general, and 3n/2 - o(n) if the rectanles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even "small" factor improvements of 4/3 or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.

    Original languageEnglish (US)
    Pages (from-to)252-267
    Number of pages16
    JournalSIAM Journal on Discrete Mathematics
    Volume15
    Issue number2
    DOIs
    StatePublished - Feb 1 2002

    Fingerprint

    Binary Space Partition
    Tiling
    Rectangle
    Partitioning
    Binary
    Upper bound
    Computational Geometry
    Tile
    Approximation Algorithms
    Upper and Lower Bounds
    Data Structures
    Lower bound

    All Science Journal Classification (ASJC) codes

    • Mathematics(all)

    Cite this

    Berman, Piotr ; Dasgupta, Bhaskar ; Muthukrishnan, S. / Exact size of binary space partitionings and improved rectangle tiling algorithms. In: SIAM Journal on Discrete Mathematics. 2002 ; Vol. 15, No. 2. pp. 252-267.
    @article{e1352c892305482cad416850aba1ad01,
    title = "Exact size of binary space partitionings and improved rectangle tiling algorithms",
    abstract = "We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n - 1 in general, and an upper bound of 2n - 1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Windmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255-259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. • A worst-case lower bound of 2n - o(n) in general, and 3n/2 - o(n) if the rectanles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even {"}small{"} factor improvements of 4/3 or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.",
    author = "Piotr Berman and Bhaskar Dasgupta and S. Muthukrishnan",
    year = "2002",
    month = "2",
    day = "1",
    doi = "10.1137/S0895480101384347",
    language = "English (US)",
    volume = "15",
    pages = "252--267",
    journal = "SIAM Journal on Discrete Mathematics",
    issn = "0895-4801",
    publisher = "Society for Industrial and Applied Mathematics Publications",
    number = "2",

    }

    Exact size of binary space partitionings and improved rectangle tiling algorithms. / Berman, Piotr; Dasgupta, Bhaskar; Muthukrishnan, S.

    In: SIAM Journal on Discrete Mathematics, Vol. 15, No. 2, 01.02.2002, p. 252-267.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Exact size of binary space partitionings and improved rectangle tiling algorithms

    AU - Berman, Piotr

    AU - Dasgupta, Bhaskar

    AU - Muthukrishnan, S.

    PY - 2002/2/1

    Y1 - 2002/2/1

    N2 - We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n - 1 in general, and an upper bound of 2n - 1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Windmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255-259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. • A worst-case lower bound of 2n - o(n) in general, and 3n/2 - o(n) if the rectanles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even "small" factor improvements of 4/3 or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.

    AB - We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: • An upper bound of 3n - 1 in general, and an upper bound of 2n - 1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Windmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255-259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. • A worst-case lower bound of 2n - o(n) in general, and 3n/2 - o(n) if the rectanles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even "small" factor improvements of 4/3 or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.

    UR - http://www.scopus.com/inward/record.url?scp=0036487459&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0036487459&partnerID=8YFLogxK

    U2 - 10.1137/S0895480101384347

    DO - 10.1137/S0895480101384347

    M3 - Article

    AN - SCOPUS:0036487459

    VL - 15

    SP - 252

    EP - 267

    JO - SIAM Journal on Discrete Mathematics

    JF - SIAM Journal on Discrete Mathematics

    SN - 0895-4801

    IS - 2

    ER -