### Abstract

This paper introduces the concept of precision-sensitive algorithms, analogous to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of O(2
^{-r}) in a time polynomial in r and 1/δ, where δ denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precision-sensitive algorithm, then we can show that unambiguous SAT (USAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.

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

Pages (from-to) | 1577-1595 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 29 |

Issue number | 5 |

DOIs | |

State | Published - Mar 2000 |

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### ASJC Scopus subject areas

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

### Cite this

*SIAM Journal on Computing*,

*29*(5), 1577-1595. https://doi.org/10.1137/S0097539798340205

**Precision-sensitive Euclidean shortest path in 3-space.** / Sellen, Jürgen; Choi, Joonsoo; Yap, Chee.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 29, no. 5, pp. 1577-1595. https://doi.org/10.1137/S0097539798340205

}

TY - JOUR

T1 - Precision-sensitive Euclidean shortest path in 3-space

AU - Sellen, Jürgen

AU - Choi, Joonsoo

AU - Yap, Chee

PY - 2000/3

Y1 - 2000/3

N2 - This paper introduces the concept of precision-sensitive algorithms, analogous to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of O(2 -r) in a time polynomial in r and 1/δ, where δ denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precision-sensitive algorithm, then we can show that unambiguous SAT (USAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.

AB - This paper introduces the concept of precision-sensitive algorithms, analogous to the well-known output-sensitive algorithms. We exploit this idea in studying the complexity of the 3-dimensional Euclidean shortest path problem. Specifically, we analyze an incremental approximation approach and show that this approach yields an asymptotic improvement of running time. By using an optimization technique to improve paths on fixed edge sequences, we modify this algorithm to guarantee a relative error of O(2 -r) in a time polynomial in r and 1/δ, where δ denotes the relative difference in path length between the shortest and the second shortest path. Our result is the best possible in some sense: if we have a strongly precision-sensitive algorithm, then we can show that unambiguous SAT (USAT) is in polynomial time, which is widely conjectured to be unlikely. Finally, we discuss the practicability of this approach. Experimental results are provided.

UR - http://www.scopus.com/inward/record.url?scp=0034538282&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034538282&partnerID=8YFLogxK

U2 - 10.1137/S0097539798340205

DO - 10.1137/S0097539798340205

M3 - Article

VL - 29

SP - 1577

EP - 1595

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -