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Unimodal-Cost k-Median on a Line

Authors Yike Chen , Chao Xu

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • unimodal-cost k-median
  • Monge
  • k-link path
  • dynamic programming
  • total monotonicity

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Abstract

Given n piecewise-linear unimodal functions f_1,… ,f_n:ℝ → ℝ and an integer 1 ≤ k ≤ n, the Unimodal-Cost k-Median problem asks for k real numbers y_1,… ,y_k minimizing ∑_{i=1}^n min_{1 ≤ r ≤ k} f_i(y_r). Let m be the number of breakpoints: the total number of affine-piece endpoint occurrences plus one occurrence at a chosen minimizer of each function. We give an exact algorithm running in O((m+nlog n)log m ⋅ min{k, log m√{klog m}, log m⋅ 2^O(√{log k log log m})}) . The first term inside the minimum comes from a direct k-stage dynamic program. The other two use the minimum-weight k-link path algorithms of Aggarwal et al. [Aggarwal et al., 1994] and Schieber [Schieber, 1998] for Monge costs, replacing their O(1) edge-weight access by batched access to the implicit transition costs.

Cite As Get BibTex

Yike Chen and Chao Xu. Unimodal-Cost k-Median on a Line. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.79

Author Details

Yike Chen
  • University of Electronic Science and Technology of China, Chengdu, China
Chao Xu
  • University of Electronic Science and Technology of China, Chengdu, China

Funding

Supported by the National Natural Science Foundation of China under grant 62372093 and by the CCF-Huawei Populus Grove Fund - Theoretical Computer Science Special Project CCF-HuaweiLK2025003.

Acknowledgements

We thank Da Wei Zheng, who came up with the variation of the algorithm for the k = 2 case and contributed to initial discussions for higher k.

References

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