## Abstract

Consider a set of n stochastic lines in R^{2}, where the existence probability of each line is determined by a fixed probability distribution. For a fixed x-coordinate q, the n lines from top to bottom can be represented by an ordered n-element sequence. Consider all the (nk) k-element sub-sequences of that n-element sequence. Each k-element sub-sequence has an associated likelihood to be the true k-topmost lines at x-coordinate q, and the one with the largest probability is defined as the most likely k-topmost lines at q. This paper studies the most likely k-topmost lines of the arrangement of n lines taken over all the x-coordinates. Let cnt be the total number of distinct sequences of the most likely k-topmost lines over all x-coordinates. The main result established is that the expected value of cnt is O(kn), which implies that it is possible to store all the distinct most likely k-topmost lines in O(k^{2}n) expected space. An example is given showing that cnt, in the worst case, can be Θ(n^{2}) even when k=1. This highlights the value of the expected bound. An algorithm is also given to compute the most likely k-topmost lines of the arrangement. Applications of this result to the stochastic Voronoi Diagram in R^{1} and to the stochastic preference top-k query in R^{2} are discussed.

Original language | English (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Discrete Algorithms |

Volume | 44 |

DOIs | |

State | Published - May 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier B.V.

## Keywords

- Algorithms
- Computational geometry
- Data structures
- Line arrangements
- Stochastic problems
- Voronoi Diagrams