### Abstract

Given a planar polygon (or chain) with a list of edges {e_{1}, e_{2}, e_{3}, ..., e_{n-1}, e_{n}}, 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(n^{2}) 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 language | English (US) |
---|---|

Pages (from-to) | 87-100 |

Number of pages | 14 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 21 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2011 |

### Fingerprint

### 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

*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.

Research output: Contribution to journal › Article

*International Journal of Computational Geometry and Applications*, vol. 21, no. 1, pp. 87-100. https://doi.org/10.1142/S0218195911003561

}

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 -