### Abstract

We consider three problems about the illumination of planar regions with floodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle φ ≤ π, n points p_{1} . . . . . p_{n} in the plane and n angles α_{1} . . . . . α_{n} such that ∑^{n}_{i=1} α_{i} ≤ θ, decide whether W can be illuminated by floodlights of angles α_{1} , . . . , α_{n} placed in some order at the points p_{1} , . . . , p_{n} and then rotated appropriately. We show that this problem is the exponential time and a specialized version of it (when φ = θ) is in NP. The second problem arises when the n points are in the complementary wedge of W and θ ≥ φ. Boss et al. have shown that a solution exists and gave an O(n log n) algorithm to place the floodlights. Here we give a matching lower bound. Problem 3 involves the illumination of the whole plane. The algorithm of Bose et al. uses an O(n log n) tripartitioning algorithm to reduce problem 3 to problem 2. We give a linear time tripartitioning algorithm of independent interest.

Original language | English (US) |
---|---|

Pages (from-to) | 57-70 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 10 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

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### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*10*(1), 57-70. https://doi.org/10.1016/S0925-7721(97)00027-8

}

*Computational Geometry: Theory and Applications*, vol. 10, no. 1, pp. 57-70. https://doi.org/10.1016/S0925-7721(97)00027-8

**Illumination by floodlights.** / Steiger, William; Streinu, Ileana.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Illumination by floodlights

AU - Steiger, William

AU - Streinu, Ileana

PY - 1998/1/1

Y1 - 1998/1/1

N2 - We consider three problems about the illumination of planar regions with floodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle φ ≤ π, n points p1 . . . . . pn in the plane and n angles α1 . . . . . αn such that ∑ni=1 αi ≤ θ, decide whether W can be illuminated by floodlights of angles α1 , . . . , αn placed in some order at the points p1 , . . . , pn and then rotated appropriately. We show that this problem is the exponential time and a specialized version of it (when φ = θ) is in NP. The second problem arises when the n points are in the complementary wedge of W and θ ≥ φ. Boss et al. have shown that a solution exists and gave an O(n log n) algorithm to place the floodlights. Here we give a matching lower bound. Problem 3 involves the illumination of the whole plane. The algorithm of Bose et al. uses an O(n log n) tripartitioning algorithm to reduce problem 3 to problem 2. We give a linear time tripartitioning algorithm of independent interest.

AB - We consider three problems about the illumination of planar regions with floodlights of prescribed angles. Problem 1 is the decision problem: given a wedge W of angle φ ≤ π, n points p1 . . . . . pn in the plane and n angles α1 . . . . . αn such that ∑ni=1 αi ≤ θ, decide whether W can be illuminated by floodlights of angles α1 , . . . , αn placed in some order at the points p1 , . . . , pn and then rotated appropriately. We show that this problem is the exponential time and a specialized version of it (when φ = θ) is in NP. The second problem arises when the n points are in the complementary wedge of W and θ ≥ φ. Boss et al. have shown that a solution exists and gave an O(n log n) algorithm to place the floodlights. Here we give a matching lower bound. Problem 3 involves the illumination of the whole plane. The algorithm of Bose et al. uses an O(n log n) tripartitioning algorithm to reduce problem 3 to problem 2. We give a linear time tripartitioning algorithm of independent interest.

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UR - http://www.scopus.com/inward/citedby.url?scp=0041125889&partnerID=8YFLogxK

U2 - 10.1016/S0925-7721(97)00027-8

DO - 10.1016/S0925-7721(97)00027-8

M3 - Article

AN - SCOPUS:0041125889

VL - 10

SP - 57

EP - 70

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 1

ER -