### Abstract

We consider the two-point query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s,t), of an Euclidean shortest obstacle-avoiding path, π(s, t), from s to t. Additionally, our data structure should allow one to report the path π(s, t), in time proportional to its (combinatorial) size. We present various methods for solving this two-point query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log^{2} n) or optimal O(log n) query times, using polynomial-space data structures, with various tradeoffs between space and query time. While several results have been known for approximate two-point Euclidean shortest path queries, it has been a well-publicized open problem to obtain sublinear query time for the exact version of the problem. Our methods also yield data structures for two-point shortest path queries on nonconvex polyhedral surfaces.

Original language | English (US) |
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Pages | 215-224 |

Number of pages | 10 |

State | Published - 1999 |

Event | Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms - Baltimore, MD, USA Duration: Jan 17 1999 → Jan 19 1999 |

### Other

Other | Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms |
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City | Baltimore, MD, USA |

Period | 1/17/99 → 1/19/99 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

*Two-point Euclidean shortest path queries in the plane*. 215-224. Paper presented at Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, USA, .