Abstract
We consider the inverse Voronoi diagram problem in trees: given a tree T with positive edge-lengths and a collection \(\mathbb {U}\) of subsets of vertices of V(T), decide whether \({\mathbb {U}}\) is a Voronoi diagram in T with respect to the shortest-path metric. We show that the problem can be solved in \(O(N+n \log ^2 n)\) time, where n is the number of vertices in T and \(N=n+\sum _{U\in {\mathbb {U}}}|U|\) is the size of the description of the input. We also provide a lower bound of \(\Omega (n \log n)\) time for trees with n vertices.
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Notes
Without the replacement \(S_\ell\) with \(S_\ell \cap W_\ell\), the lemma is actually not true because it can happen that \(s_\ell \in U_1\cap U_\ell\). Indeed, we could have \(s_\ell \in S_\ell \cap U_\ell \cap U_1\), which is not a valid placement in I but would be a valid placement in \(I'\).
In our application, the operation IntContaining(y) is used only internally, but it seems useful in general and we keep it in this description of external operations.
In the process we destroy the data structures for \({{\mathbb {I}}}(A(v_2))\) and \({{\mathbb {I}}}(B(v_2))\).
References
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Acknowledgements
We are very grateful to the anonymous reviewers for pointing out an error in the previous version of Sect. 3.2 and several other useful corrections. Part of this work was done at the 21st Korean Workshop on Computational Geometry, held in Rogla, Slovenia, in June 2018. We thank all workshop participants for their helpful comments.
Funding
Édouard Bonnet is supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Sergio Cabello is supported by the Slovenian Research Agency, Program P1-0297 and Projects J1-8130, J1-8155, J1-9109, J1-1693, J1-2452. Bojan Mohar is on leave from IMFM & FMF, Department of Mathematics, University of Ljubljana and is supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Projects J1-8130 and J1-2452 of ARRS (Slovenia). Hebert Pérez-Rosés is partially supported by Grant MTM2017-86767-R from the Spanish Ministry of Economy, Industry and Competitiveness.
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Bonnet, É., Cabello, S., Mohar, B. et al. The Inverse Voronoi Problem in Graphs II: Trees. Algorithmica 83, 1165–1200 (2021). https://doi.org/10.1007/s00453-020-00774-8
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DOI: https://doi.org/10.1007/s00453-020-00774-8