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Reachability Problems for Transmission Graphs

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Abstract

Let P be a set of n points in the plane where each point p of P is associated with a radius \(r_p>0\). The transmission graph \(G=(P,E)\) of P is defined as the directed graph such that E contains an edge from p to q if and only if \(|pq|\le r_p\) for any two points p and q in P, where |pq| denotes the Euclidean distance between p and q. In this paper, we present a data structure of size \(O(n^{5/3})\) such that for any two points in P, we can check in \(O(n^{2/3})\) time if there is a path in G between the two points. This is the first data structure for answering reachability queries whose performance depends only on n but not on the ratio between the largest and smallest radii.

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Notes

  1. Kaplan et al. mentioned that this algorithm takes an \(O(n\log ^5 n)\) time. However, this can be improved automatically into \(O(n\log ^4 n)\) using a data structure of [4].

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  1. Pohang University of Science and Technology, Pohang, Korea

    Shinwoo An & Eunjin Oh

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  1. Shinwoo An
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  2. Eunjin Oh
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Correspondence to Eunjin Oh.

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A preliminary version of this article has appeared in the proceedings of WADS 2021.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1012742).

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An, S., Oh, E. Reachability Problems for Transmission Graphs. Algorithmica 84, 2820–2841 (2022). https://doi.org/10.1007/s00453-022-00985-1

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