### Abstract

An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α > 0 and all its edges are at least β.diam(c)-long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of "constant description complexity" is O(λ _{s+2}(n) log ^{2} n log log n), where s is the maximum number of intersections between the boundaries of any pair of given objects, and λ _{s} (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence. Our result extends and improves previous results concerning convex α-fat objects.

Original language | English (US) |
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Pages (from-to) | 775-787 |

Number of pages | 13 |

Journal | SIAM Journal on Computing |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Fat objects
- Motion planning
- Realistic input models

### ASJC Scopus subject areas

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

### Cite this

**The complexity of the union of (α,β)-covered objects.** / Efrat, Alon.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 34, no. 4, pp. 775-787. https://doi.org/10.1137/S0097539702407515

}

TY - JOUR

T1 - The complexity of the union of (α,β)-covered objects

AU - Efrat, Alon

PY - 2005

Y1 - 2005

N2 - An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α > 0 and all its edges are at least β.diam(c)-long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of "constant description complexity" is O(λ s+2(n) log 2 n log log n), where s is the maximum number of intersections between the boundaries of any pair of given objects, and λ s (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence. Our result extends and improves previous results concerning convex α-fat objects.

AB - An (α, β)-covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α > 0 and all its edges are at least β.diam(c)-long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)-covered objects of "constant description complexity" is O(λ s+2(n) log 2 n log log n), where s is the maximum number of intersections between the boundaries of any pair of given objects, and λ s (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence. Our result extends and improves previous results concerning convex α-fat objects.

KW - Fat objects

KW - Motion planning

KW - Realistic input models

UR - http://www.scopus.com/inward/record.url?scp=24344452560&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24344452560&partnerID=8YFLogxK

U2 - 10.1137/S0097539702407515

DO - 10.1137/S0097539702407515

M3 - Article

VL - 34

SP - 775

EP - 787

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -