## Touring a Sequence of Polygons (2003)

Venue: | In Proc. 35th Annu. ACM Sympos. Theory Comput |

Citations: | 31 - 4 self |

### BibTeX

@INPROCEEDINGS{Dror03touringa,

author = {Moshe Dror and Alon Efrat and Anna Lubiw and Joseph S. B. Mitchell},

title = {Touring a Sequence of Polygons},

booktitle = {In Proc. 35th Annu. ACM Sympos. Theory Comput},

year = {2003},

pages = {473--482},

publisher = {ACM Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NP-hard.

### Citations

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(Show Context)
Citation Context ...orientations (axis-parallel, plus edges of angle 45 degrees) instead of just the two axis-parallel orientations, the problem becomes hard. The proof is based on a careful adaptation of the Canny-Reif =-=[4]-=- proof of NPhardness of the three-dimensional shortest path problem, using some new gadgets. Theorem 6. The touring polygons problem is NP-hard, for any Lp metric (p # 1), in the case of nonconvex pol... |

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Citation Context ... P i 's. If the order in which the polygons P i must be visited is not specified, then the touring polygons problem becomes the Traveling Salesperson Problem with Neighborhoods, which is NP-hard. See =-=[20]-=-. The touring polygons problem can be modeled as a special kind of 3-dimensional shortest path problem among polyhedral obstacles. Imagine k very large sheets of paper stacked up in parallel planes or... |

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Citation Context ... of S R i . We preprocess S A i-1 and S R i for point location queries. Constructing S F i . We preprocess F i for ray shooting queries; this takes O(|F i |) time, and allows queries in O(log n) time =-=[17]-=-. We call Query(v, S R i ), for each reflex vertex v of F i , to find the last edge l of a shortest fenced i-path to v ignoring the fence F i . Using a ray shooting query in F i , we determine if l is... |

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Citation Context ...as the three-dimensional shortest path problem among obstacles. It is also "just as easy", in that known techniques to obtain an approximately shortest path among obstacles in three dimensio=-=ns (e.g., [1, 10, 11, 12, 22]) apply to-=- the TPP Figure 5: "Ganging" three shu#e gadgets in parallel. b = most significant bit 2 shortest path classes n i-1 shufflers (n-i+1) shufflers 1 3 4 6 1 3 4 6 4 5 2 3 3-way splitter 3-way ... |

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Citation Context ...the starting point of the tour; the "floating" WRP, with no point s specified, requires time O(n) times that of the fixed WRP, as shown recently by Tan [26]. The WRP was introduced by Chin a=-=nd Ntafos [9, 7]-=-, who claimed in [7] an O(n 4 ) time algorithm for the fixed WRP in a simple polygon. However, years later, there was a flaw discovered in their algorithm, and several attempts were made to correct it... |

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Citation Context ...as the three-dimensional shortest path problem among obstacles. It is also "just as easy", in that known techniques to obtain an approximately shortest path among obstacles in three dimensio=-=ns (e.g., [1, 10, 11, 12, 22]) apply to-=- the TPP Figure 5: "Ganging" three shu#e gadgets in parallel. b = most significant bit 2 shortest path classes n i-1 shufflers (n-i+1) shufflers 1 3 4 6 1 3 4 6 4 5 2 3 3-way splitter 3-way ... |

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Citation Context ...the starting point of the tour; the "floating" WRP, with no point s specified, requires time O(n) times that of the fixed WRP, as shown recently by Tan [26]. The WRP was introduced by Chin a=-=nd Ntafos [9, 7]-=-, who claimed in [7] an O(n 4 ) time algorithm for the fixed WRP in a simple polygon. However, years later, there was a flaw discovered in their algorithm, and several attempts were made to correct it... |

36 |
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(Show Context)
Citation Context ...blems seem di#erent from the TPP in that no ordering of the P i 's has been specified, it is easy to argue that a shortest tour must visit the P i 's in the same order as they meet the boundary of P (=-=[8, 21]-=-). The zookeeper problem, introduced by Chin and Ntafos [8], has an O(n log n) time algorithm [3] utilizing the full shortest path map. The safari problem was introduced by Ntafos [21], who claimed an... |

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Citation Context ...as the three-dimensional shortest path problem among obstacles. It is also "just as easy", in that known techniques to obtain an approximately shortest path among obstacles in three dimensio=-=ns (e.g., [1, 10, 11, 12, 22]) apply to-=- the TPP Figure 5: "Ganging" three shu#e gadgets in parallel. b = most significant bit 2 shortest path classes n i-1 shufflers (n-i+1) shufflers 1 3 4 6 1 3 4 6 4 5 2 3 3-way splitter 3-way ... |

19 |
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(Show Context)
Citation Context ...r which our results imply an O(n 3 log n) algorithm (improving upon O(n 4 )). (Our results imply also an improved bound -- O(n 4 log n) instead of O(n 5 ) -- for the "floating" watchman rout=-=e problem [26]-=-.) One of the main significances of our new method is not just that it yields a substantially faster algorithm for the watchman route problem, but also that it avoids using dynamic programming and com... |

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11 |
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(Show Context)
Citation Context ...blems seem di#erent from the TPP in that no ordering of the P i 's has been specified, it is easy to argue that a shortest tour must visit the P i 's in the same order as they meet the boundary of P (=-=[8, 21]-=-). The zookeeper problem, introduced by Chin and Ntafos [8], has an O(n log n) time algorithm [3] utilizing the full shortest path map. The safari problem was introduced by Ntafos [21], who claimed an... |

11 |
Finding shortest safari routes in simple polygons
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(Show Context)
Citation Context ... has an O(n log n) time algorithm [3] utilizing the full shortest path map. The safari problem was introduced by Ntafos [21], who claimed an O(n 3 ) time algorithm, which was then improved to O(n 2 ) =-=[24]-=-. However, Tan and Hirata [28] found an error in the earlier analysis and presented the current best algorithm with running time O(n 3 ). Our algorithm solves a much more general problem, and improves... |

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(Show Context)
Citation Context |

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7 | Shortest zookeeper routes in simple polygons - Tan |

6 | An O(n log n) algorithm for the zoo-keepers problem
- Bespamyatnikh
(Show Context)
Citation Context ...argue that a shortest tour must visit the P i 's in the same order as they meet the boundary of P ([8, 21]). The zookeeper problem, introduced by Chin and Ntafos [8], has an O(n log n) time algorithm =-=[3]-=- utilizing the full shortest path map. The safari problem was introduced by Ntafos [21], who claimed an O(n 3 ) time algorithm, which was then improved to O(n 2 ) [24]. However, Tan and Hirata [28] fo... |

2 |
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(Show Context)
Citation Context ...ygons must be visited. (Certain applications specify the order in which parts must be cut; without a given order, the problem is obviously NP-hard.) This problem is known as the parts cutting problem =-=[16]-=-. The parts cutting problem is exactly the unconstrained TPP with disjoint P i 's and F i = R 2 , #i. Our algorithm solves the parts cutting problem in time O(kn log(n/k)); see Section 3. 2.2 The Safa... |

2 |
Corrigendum to "An Incremental Algorithm for Constructing Shortest Watchman Routes
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(Show Context)
Citation Context ...ade to correct it (some of which were themselves flawed). The best current algorithm for the fixed WRP, based on a relatively complex dynamic programming algorithm, is due to Tan, Hirata, and Inagaki =-=[25]-=- and runs in time O(n 4 ). Our results imply a new, simpler algorithm for the fixed WRP that runs in time O(n 3 log n). In order to see that the WRP is a special case of the TPP, we recall the notion ... |