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Maximizing Dominance in the Plane and its Applications

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Abstract

Given a set P of n weighted points, a set Q of m points in the plane, and a positive integer k, we consider the optimization problem of finding a subset of Q with at most k points that dominates a subset of P with maximum total weight. A set \(Q'\) of points in the plane dominates a point p in the plane if some point \(q\in Q'\) satisfies \(x(p)\leqslant x(q)\) and \(y(p)\leqslant y(q)\). We present an efficient algorithm solving this problem in \(O(k(n+m)\log m)\) time and \(O(n+m)\) space. Our result implies algorithms with better time bounds for related problems, including the disjoint union of cliques problem for interval graphs (equivalently, the hitting intervals problem) and the top-k representative skyline points problem in the plane.

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Notes

  1. Given the amount of interrelated results and possible names for the problem, we hope we are not victims of the same problem.

  2. A data structure with similar guarantees can be constructed using cut-link trees [16], top trees [2], or other dynamic trees. Depending on the actual dynamic tree underlying the construction, the bounds may be amortized.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

    Jongmin Choi

  2. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sergio Cabello

  3. IMFM, Ljubljana, Slovenia

    Sergio Cabello

  4. Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, South Korea

    Hee-Kap Ahn

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  1. Jongmin Choi
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  2. Sergio Cabello
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  3. Hee-Kap Ahn
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Correspondence to Hee-Kap Ahn.

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Work by Choi and Ahn was partly supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)) and (No. 2019-0-01906, Artificial Intelligence Graduate School Program(POSTECH)). Work by Cabello was supported by the Slovenian Research Agency, program P1-0297 and projects J1-8130, J1-8155, J1-9109, J1-1693, J1-2452.

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Choi, J., Cabello, S. & Ahn, HK. Maximizing Dominance in the Plane and its Applications. Algorithmica 83, 3491–3513 (2021). https://doi.org/10.1007/s00453-021-00863-2

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