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Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

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Abstract

We present the first sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3), which is a significant improvement over the previous algorithms taking time O(n).

We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7, where D is the diameter of the polygon.

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Authors and Affiliations

  1. Institute of Informatics, Warsaw University, Warsaw, Poland

    Jakub Pawlewicz

  2. MIT, Cambridge, MA, USA

    Mihai Pătraşcu

Authors
  1. Jakub Pawlewicz
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  2. Mihai Pătraşcu
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Correspondence to Jakub Pawlewicz.

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This work represents a merging of 19 and 21, with additional extensions.

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Pawlewicz, J., Pătraşcu, M. Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons. Algorithmica 55, 271–282 (2009). https://doi.org/10.1007/s00453-008-9221-z

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  • DOI: https://doi.org/10.1007/s00453-008-9221-z

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