### Abstract

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be "locally" random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley's theory of ⊖-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson-Shor type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to average-case analysis in computational geometry.

Original language | English (US) |
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Title of host publication | Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08 |

Pages | 156-163 |

Number of pages | 8 |

DOIs | |

State | Published - Dec 12 2008 |

Event | 24th Annual Symposium on Computational Geometry, SCG'08 - College Park, MD, United States Duration: Jun 9 2008 → Jun 11 2008 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Other

Other | 24th Annual Symposium on Computational Geometry, SCG'08 |
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Country | United States |

City | College Park, MD |

Period | 6/9/08 → 6/11/08 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Keywords

- Clarkson-shor bound
- Expander graphs
- Randomized incremental constructions

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## Cite this

*Proceedings of the 24th Annual Symposium on Computational Geometry 2008, SCG'08*(pp. 156-163). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1377676.1377701