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Sampling and sparsification for approximating the packedness of trajectories and detecting gatherings

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Abstract

Packedness is a measure defined for curves as the ratio of maximum curve length inside any disk divided by its radius. Sparsification allows us to reduce the number of candidate disks for maximum packedness to a polynomial amount in terms of the number of vertices of the polygonal curve. This gives an exact algorithm for computing packedness. We prove that using a fat shape, such as a square, instead of a disk gives a constant factor approximation for packedness. Further sparsification using well-separated pair decomposition improves the time complexity at the cost of losing some accuracy. By adjusting the ratio of the separation factor and the size of the query, we improve the approximation factor of the existing algorithm for packedness using square queries. Our experiments show that uniform sampling works well for finding the average packedness of trajectories with almost constant speed. The empirical results confirm that the sparsification method approximates the maximum packedness for arbitrary polygonal curves. In big data models such as massively parallel computations, both sampling and sparsification are efficient and take a constant number of rounds. Most existing algorithms use line-sweeping which is sequential in nature. Also, we design two data-structures for computing the length of the curve inside a query shape: an exact data-structure for disks called hierarchical aggregated queries and an approximate data-structure for a given set of square queries. Using our modified segment tree, we achieve a near-linear time approximation algorithm.

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Funding

V. Keikha was Supported by the Czech Science Foundation, Grant No. GJ19-06792Y, and with institutional support RVO:67985807.

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Authors and Affiliations

  1. Department of Computer Engineering, Sharif University of Technology, Azadi Ave, Tehran, Iran

    Sepideh Aghamolaei

  2. The Czech Academy of Sciences, Institute of Computer Science, Pod Vodárenskou věží 2, 182 07, Prague, Czech Republic

    Vahideh Keikha

  3. Department of Computer Engineering, Sharif University of Technology, Tehran, Iran

    Mohammad Ghodsi

  4. Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

    Ali Mohades

Authors
  1. Sepideh Aghamolaei
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  2. Vahideh Keikha
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  3. Mohammad Ghodsi
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  4. Ali Mohades
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Contributions

Sepideh Aghamolaei, Vahideh Keikha, Mohammad Ghodsi, and Ali Mohades contributed to writing, review, and editing. Sepideh Aghamolaei contributed to implementation. Mohammad Ghodsi and Ali Mohades contributed to supervision.

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Correspondence to Sepideh Aghamolaei.

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Publicly available at https://github.com/saghamolaei/Packedness.

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Aghamolaei, S., Keikha, V., Ghodsi, M. et al. Sampling and sparsification for approximating the packedness of trajectories and detecting gatherings. Int J Data Sci Anal 15, 201–216 (2023). https://doi.org/10.1007/s41060-021-00301-0

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  • DOI: https://doi.org/10.1007/s41060-021-00301-0

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