Computing signed permutations of polygons

Greg Aloupis, Prosenjit Bose, Erik D. Demaine, Stefan Langerman, Henk Meijer, Mark Overmars, Godfried Toussaint

    Research output: Contribution to journalArticle

    Abstract

    Given a planar polygon (or chain) with a list of edges {e1, e2, e3, ..., en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

    Original languageEnglish (US)
    Pages (from-to)87-100
    Number of pages14
    JournalInternational Journal of Computational Geometry and Applications
    Volume21
    Issue number1
    DOIs
    StatePublished - Feb 1 2011

    Fingerprint

    Signed Permutations
    Polygon
    Stars
    Mirrors
    Computing
    Transposition
    Swap
    Signed
    Reversal
    Mirror Symmetry
    Rotational symmetry
    Rectangle
    Consecutive
    Simplicity
    Star
    NP-complete problem

    Keywords

    • Computational geometry
    • geometric permutation
    • polygonal reconfiguration

    ASJC Scopus subject areas

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

    Cite this

    Aloupis, G., Bose, P., Demaine, E. D., Langerman, S., Meijer, H., Overmars, M., & Toussaint, G. (2011). Computing signed permutations of polygons. International Journal of Computational Geometry and Applications, 21(1), 87-100. https://doi.org/10.1142/S0218195911003561

    Computing signed permutations of polygons. / Aloupis, Greg; Bose, Prosenjit; Demaine, Erik D.; Langerman, Stefan; Meijer, Henk; Overmars, Mark; Toussaint, Godfried.

    In: International Journal of Computational Geometry and Applications, Vol. 21, No. 1, 01.02.2011, p. 87-100.

    Research output: Contribution to journalArticle

    Aloupis, G, Bose, P, Demaine, ED, Langerman, S, Meijer, H, Overmars, M & Toussaint, G 2011, 'Computing signed permutations of polygons', International Journal of Computational Geometry and Applications, vol. 21, no. 1, pp. 87-100. https://doi.org/10.1142/S0218195911003561
    Aloupis G, Bose P, Demaine ED, Langerman S, Meijer H, Overmars M et al. Computing signed permutations of polygons. International Journal of Computational Geometry and Applications. 2011 Feb 1;21(1):87-100. https://doi.org/10.1142/S0218195911003561
    Aloupis, Greg ; Bose, Prosenjit ; Demaine, Erik D. ; Langerman, Stefan ; Meijer, Henk ; Overmars, Mark ; Toussaint, Godfried. / Computing signed permutations of polygons. In: International Journal of Computational Geometry and Applications. 2011 ; Vol. 21, No. 1. pp. 87-100.
    @article{706fb1d658274165b0a1a43db6e0c411,
    title = "Computing signed permutations of polygons",
    abstract = "Given a planar polygon (or chain) with a list of edges {e1, e2, e3, ..., en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.",
    keywords = "Computational geometry, geometric permutation, polygonal reconfiguration",
    author = "Greg Aloupis and Prosenjit Bose and Demaine, {Erik D.} and Stefan Langerman and Henk Meijer and Mark Overmars and Godfried Toussaint",
    year = "2011",
    month = "2",
    day = "1",
    doi = "10.1142/S0218195911003561",
    language = "English (US)",
    volume = "21",
    pages = "87--100",
    journal = "International Journal of Computational Geometry and Applications",
    issn = "0218-1959",
    publisher = "World Scientific Publishing Co. Pte Ltd",
    number = "1",

    }

    TY - JOUR

    T1 - Computing signed permutations of polygons

    AU - Aloupis, Greg

    AU - Bose, Prosenjit

    AU - Demaine, Erik D.

    AU - Langerman, Stefan

    AU - Meijer, Henk

    AU - Overmars, Mark

    AU - Toussaint, Godfried

    PY - 2011/2/1

    Y1 - 2011/2/1

    N2 - Given a planar polygon (or chain) with a list of edges {e1, e2, e3, ..., en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

    AB - Given a planar polygon (or chain) with a list of edges {e1, e2, e3, ..., en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.

    KW - Computational geometry

    KW - geometric permutation

    KW - polygonal reconfiguration

    UR - http://www.scopus.com/inward/record.url?scp=79952595547&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=79952595547&partnerID=8YFLogxK

    U2 - 10.1142/S0218195911003561

    DO - 10.1142/S0218195911003561

    M3 - Article

    VL - 21

    SP - 87

    EP - 100

    JO - International Journal of Computational Geometry and Applications

    JF - International Journal of Computational Geometry and Applications

    SN - 0218-1959

    IS - 1

    ER -